APPLICATION OF RIGOROUS HIGH-ORDER METHODS AND NORMAL FORMS TO NONLINEAR SYSTEMS By Adrian Weisskopf A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Physics โ€“ Doctor of Philosophy 2021 ABSTRACT APPLICATION OF RIGOROUS HIGH-ORDER METHODS AND NORMAL FORMS TO NONLINEAR SYSTEMS By Adrian Weisskopf The nonlinearities of dynamical systems often display the most interesting and fascinating behavior. At the same time, those nonlinearities complicate finding closed form analytic solutions, especially for complex systems, to the point where it is often impossible. Differential algebra (DA) based methods allow us to analyze those systems with all their nonlinearities up to arbitrary order in an automated, computer based framework that operates with floating point accuracy. This thesis will investigate repetitive dynamical systems from seemingly unrelated fields of study using DA methods such as DA based transfer and Poincarรฉ maps, the DA normal form algorithm, normal form defect studies, and verified methods based on Taylor Models. The common mathematical underpinnings of those dynamical systems allow us to analyze them with different techniques that have the same methods at their core. Specifically, we will analyze resonances, associated fixed point structures, and oscillation periods of particles in the accelerator storage ring of the Muon ๐‘”-2 Experiment at Fermilab to gain a detailed understanding of the stability of the system and the potential loss mechanism of particles. If successful, the Muon ๐‘”-2 Experiment raises existential questions about the completeness of the Standard Model of particle physics, which makes our efforts to understand the stability of the system highly relevant. The same methods used for the analysis of the accelerator storage ring will also be used to generate far reaching sets of satellite orbits for formation flying missions under the Earthโ€™s gravitational zonal perturbations. Our approach is particularly elegant and precise, and its theoretical limits are beyond the range of practical applications. One central method in both of those analyses is the DA normal form algorithm. Using the mechanical device of the centrifugal governor as an illustrative example problem, the special properties of the resulting normal form, the sensitivities and limitations of the algorithm, and its resulting quantities are explained in detail. We also will provide first results and an outlook for future work of the presented methods in the realm of verified methods, and illustrate the current possibilities as well as future opportunities and challenges. In particular, Taylor Model based verified global optimization is introduced and used to calculate rigorous stability estimates for different configurations of the Muon ๐‘”-2 Storage Ring. To my parents and grandparents. iv ACKNOWLEDGEMENTS First of all, I would like to thank my academic advisor Professor Martin Berz for his continuous support, patience, and helpful guidance not only in my research, but also during my Ph.D. time in general. I really enjoyed diving into complex problems with him and discovering the key mechanisms at play. His approach of analyzing the fundamental components of a problem strongly influenced my way of structuring problems and attempting their solution. Furthermore, I particularly appreciated the collaborative work with Roberto Armellin, David Tarazona, Kyoko Makino, and Eremey Valetov and would like to thank all of them for the insightful discussions and welcoming atmosphere on and off work. I am proud of our joint contributions to the scientific community and really enjoyed the process of getting there. I would like to additionally thank Kyoko for her remarkable support and attention to detail, which contributed to this work. I also shall not forget to thank Scott Pratt, Mark Dykman, Vladimir Zelevinsky, together with Martin and Kyoko for being so kind and agreeing to serve on my thesis committee. Last but not least, I cannot thank enough my girlfriend Franzi, my family and friends for their continuous support during my Ph.D. time. This thesis and Ph.D. was only possible due to the generous scholarship by the Studienstiftung des deutschen Volkes. Additionally, it was supported by the US Department of Energy (DOE) and the Mathematical Sciences Research Institute (MSRI) in Berkeley. This research used resources of the National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy Office of Science User Facility located at Lawrence Berkeley National Laboratory, operated under Contract No. DE-AC02-05CH11231. v TABLE OF CONTENTS LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 CHAPTER 2 METHODS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1 The Differential Algebra (DA) Framework . . . . . . . . . . . . . . . . . . . . . . 5 2.2 DA Transfer Maps and Poincarรฉ Maps . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 The DA Normal Form Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3.1 Tunes, Tune Shifts, and Normal Form Radii . . . . . . . . . . . . . . . . . 16 2.3.2 Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.4 The Normal Form Defect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.5 Verified Computations Using Taylor Models (TM) . . . . . . . . . . . . . . . . . . 22 2.5.1 Interval Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.5.2 Taylor Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.6 Taylor Model based Verified Global Optimizers . . . . . . . . . . . . . . . . . . . 27 CHAPTER 3 AN EXAMPLE-DRIVEN WALK-THROUGH OF THE DA NORMAL FORM ALGORITHM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.1 The Centrifugal Governor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.1.1 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.1.2 The Equilibrium Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.1.3 The Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.1.4 Illustration of System Dynamics . . . . . . . . . . . . . . . . . . . . . . . 35 3.2 Map Calculation via Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.3 The DA Normal Form Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.3.1 The Parameter Dependent Fixed Point . . . . . . . . . . . . . . . . . . . . 40 3.3.2 The Linear Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.3.3 The Nonlinear Transformations . . . . . . . . . . . . . . . . . . . . . . . . 44 3.3.3.1 General ๐‘šth Order Nonlinear Transformation . . . . . . . . . . . 44 3.3.3.2 Explicit Second Order Nonlinear Transformation . . . . . . . . . 46 3.3.3.3 Explicit Third Order Nonlinear Transformation . . . . . . . . . . 49 3.3.3.4 The Effect of the Second Order Transformation on Third Order Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.3.4 Transformation back to Real Space Normal Form . . . . . . . . . . . . . . 52 3.3.5 Invariant Normal Form Radius . . . . . . . . . . . . . . . . . . . . . . . . 55 3.3.6 Angle Advancement, Tune and Tune Shifts . . . . . . . . . . . . . . . . . . 56 3.4 Visualization of the Different Order Normal Forms and Conclusion . . . . . . . . . 58 CHAPTER 4 BOUNDED MOTION PROBLEM . . . . . . . . . . . . . . . . . . . . . . 61 4.1 Introduction to Bounded Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 vi 4.2 Understanding Orbital Motion Under Gravitational Perturbation . . . . . . . . . . 64 4.2.1 The Perturbed Gravitational Potential . . . . . . . . . . . . . . . . . . . . 64 4.2.2 The Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.2.3 The Kepler Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.2.4 Orbits Under Gravitational Perturbation . . . . . . . . . . . . . . . . . . . 67 4.2.5 The Bounded Motion Conditions by Xu et al. . . . . . . . . . . . . . . . . 69 4.2.6 The Fixed Point Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.3 Method of Bounded Motion Design Under Zonal Perturbation . . . . . . . . . . . 71 4.3.1 The Poincarรฉ Surface Space . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.3.2 The Fixed Point Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.3.3 The Calculation of Poincarรฉ Return Map . . . . . . . . . . . . . . . . . . . 73 4.3.4 The Normal Form Averaging . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.4 Bounded Motion Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.4.1 Bounded Motion in Low Earth Orbit . . . . . . . . . . . . . . . . . . . . . 77 4.4.2 Bounded Motion in Medium Earth Orbit . . . . . . . . . . . . . . . . . . . 82 4.4.3 Testing the Limitations of the DANF Method . . . . . . . . . . . . . . . . 86 4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 CHAPTER 5 STABILITY ANALYSIS OF MUON ๐‘”-2 STORAGE RING . . . . . . . . . 92 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.2 Storage Ring Simulation Using Poincarรฉ Maps . . . . . . . . . . . . . . . . . . . . 95 5.3 The Closed Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.3.1 The Closed Orbit Under Perturbation . . . . . . . . . . . . . . . . . . . . . 97 5.3.2 The Momentum Dependence of the Closed Orbit . . . . . . . . . . . . . . 100 5.3.3 The Relevance of Closed Orbits . . . . . . . . . . . . . . . . . . . . . . . 101 5.4 Tune Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.4.1 Tunes of the Momentum Dependent Closed Orbit . . . . . . . . . . . . . . 103 5.4.2 The Amplitude Dependent Tune Shifts . . . . . . . . . . . . . . . . . . . . 105 5.4.3 The Tune Footprint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.5 Stability and Loss Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5.5.1 The Normal Form Defect of Tracked Particles . . . . . . . . . . . . . . . . 115 5.5.2 Lost Muon Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.5.3 Period-3 Fixed Point Structures . . . . . . . . . . . . . . . . . . . . . . . . 134 5.5.4 Muon Loss Rates from Simulation . . . . . . . . . . . . . . . . . . . . . . 138 5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 CHAPTER 6 VERIFIED CALCULATIONS USING TAYLOR MODELS . . . . . . . . . 143 6.1 The Rosenbrock Optimization Problem . . . . . . . . . . . . . . . . . . . . . . . . 144 6.1.1 The Rosenbrock Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 6.1.2 Global Optimization Using COSY-GO . . . . . . . . . . . . . . . . . . . . 146 6.1.2.1 Illustration of the Cluster Effect and Dependency Problem using the 2D Rosenbrock Function . . . . . . . . . . . . . . . . . . . . 147 6.1.2.2 Performance of COSY-GO for High Dimensional Rosenbrock Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 6.2 The Lennard-Jones Potential Problem . . . . . . . . . . . . . . . . . . . . . . . . 152 vii 6.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 6.2.1.1 The Lennard-Jones Potential ๐‘ˆLJ . . . . . . . . . . . . . . . . . 153 6.2.1.2 Configurations of Particles S ๐‘˜ . . . . . . . . . . . . . . . . . . . 154 6.2.1.3 The Lennard-Jones Optimization Problem and its Challenges . . 155 6.2.2 Minimum Energy Lennard-Jones Configurations in 1D . . . . . . . . . . . 157 6.2.2.1 Coordinate System, Numbering Scheme, and Variable Defini- tion in 1D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 6.2.2.2 Upper Bounds ๐‘Ÿ ๐‘˜,UB and ๐‘ฃ ๐‘ฅ,UB on Inter-Particle Distances of Minimum Energy Configurations in 1D . . . . . . . . . . . . . . 158 6.2.2.3 The Upper Bound ๐‘ˆ ๐‘˜,UB on the Minimum Energy . . . . . . . . 159 6.2.2.4 The Lower Bound ๐‘Ÿ ๐‘˜,LB on ๐‘Ÿ๐‘– ๐‘— . . . . . . . . . . . . . . . . . . . 160 6.2.2.5 The Infinite 1D Equidistant Configuration . . . . . . . . . . . . . 161 6.2.2.6 The Verified Global Optimization Results for Configurations of ๐‘˜ Particles in 1D . . . . . . . . . . . . . . . . . . . . . . . . . . 163 6.2.2.7 The Verified Global Optimization Results for Symmetric Con- figurations of ๐‘˜ Particles in 1D . . . . . . . . . . . . . . . . . . . 169 6.2.2.8 Redundancies and Penalty Functions . . . . . . . . . . . . . . . 175 6.2.3 Minimum Energy Lennard-Jones Configurations in 2D and 3D . . . . . . . 176 6.2.3.1 Coordinate System, Numbering Scheme, and Variable Definition 176 6.2.3.2 Upper Bounds ๐‘Ÿ ๐‘˜,UB and ๐‘ฃ ๐‘ฅ,UB on Inter-Particle Distances of Minimum Energy Configurations in 2D and 3D . . . . . . . . . . 179 6.2.3.3 The Upper Bound ๐‘ˆ ๐‘˜,UB on the Minimum Energy . . . . . . . . 180 6.2.3.4 Setup of Initial Search Domain of the Optimization Variables in 2D and 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 6.2.3.5 The Evaluation of the Objective Function . . . . . . . . . . . . . 183 6.2.3.6 Taylor Model Evaluation of Piecewise Defined Functions . . . . . 184 6.2.3.7 The Verified Global Optimization Results for Configurations of ๐‘˜ Particles in 2D . . . . . . . . . . . . . . . . . . . . . . . . . . 187 6.2.3.8 The Verified Global Optimization Results for Configurations of ๐‘˜ Particles in 3D . . . . . . . . . . . . . . . . . . . . . . . . . . 197 6.2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 6.3 Verified Stability Analysis of Dynamical Systems . . . . . . . . . . . . . . . . . . 202 6.3.1 The Potential Implications for the Bounded Motion Problem . . . . . . . . 202 6.3.2 The Implications for the Stability Analysis of the Muon ๐‘”-2 Storage Ring . 204 6.3.3 The Normal Form Defect as the Objective Function for the Optimization . . 206 6.3.4 The Search Domain in the Form of Onion Layers . . . . . . . . . . . . . . 206 6.3.5 The Complexity and Nonlinearity of the Normal Form Defect Function . . 208 6.3.6 The Results of the Verified Global Optimization of the Normal Form Defect 212 6.3.7 Comparison of Verified Nekhoroshev-type Stability Estimates to Actual Rates of Divergence in Example Island Structure . . . . . . . . . . . . . . 218 6.3.8 Relevance of the ESQ Voltage on the Stability . . . . . . . . . . . . . . . . 219 6.3.9 Comparison of Nonverified and Verified Normal Form Defect Analysis . . 224 6.3.10 The Analysis of the Effect of Normal Form Transformations of Different Order on the Normal Form Defect . . . . . . . . . . . . . . . . . . . . . . 229 viii CHAPTER 7 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 ix LIST OF TABLES Table 3.1: List of stable equilibrium angles ๐œ™0 of the centrifugal governor arms for some specific rotation frequencies ๐œ”. . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 โˆš Table 3.2: Integration result for map around equilibrium state (๐œ™0 (๐œ” = 2) = ๐œ‹/3, 0) integrated until ๐‘ก = 1 using an order 20 Picard-iteration based integrator with stepsize โ„Ž = 10โˆ’3 over 1000 iterations within COSY INFINITY. The component M+0 = ๐‘„(๐‘ž 0 , ๐‘ 0 ) is on the left, Mโˆ’ 0 = ๐‘ƒ(๐‘ž 0 , ๐‘ 0 ) on the right. . . . . 39 Table 3.3: Coefficients of M1 up to order three. Note the complex conjugate property ยฑ S๐‘š(๐‘˜ โˆ“ = Sฬ„๐‘š(๐‘˜ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ,๐‘˜ ) + โˆ’ ,๐‘˜ ) โˆ’ + Table 3.4: The values of the T2(๐‘˜ยฑ ยฑ and O3(๐‘˜ . Note that T2 and O3 and therefore + ,๐‘˜ โˆ’ ) + ,๐‘˜ โˆ’ ) A2 and its inverse are real with A+๐‘š(๐‘˜ ,๐‘˜ ) = Aโˆ’ ๐‘š(๐‘˜ ,๐‘˜ ) . . . . . . . . . . . . . . 47 + โˆ’ โˆ’ + Table 3.5: New coefficients of third order of M2 after the second order transformation. Note that the first order terms remain unchanged and that the second order terms are all eliminated by the second order transformation. Interestingly, the second order transformation caused some terms of the third order to disappear in this specific case, which is not a general property of the second order transformation. The emphasized terms are surviving the third order transformation as explained in the following subsection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Table 3.6: The values of the T3(๐‘˜ ยฑ + . The values for T3(2,1) โˆ’ and T3(1,2) cannot be + ,๐‘˜ โˆ’ ) calculated because the denominator in Eq. (3.36) is zero. Note that T3(๐‘˜ + = + ,๐‘˜ โˆ’ ) โˆ’ T3(๐‘˜ ,๐‘˜ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 โˆ’ + Table 3.7: The normal form map MNF up to order three. The component M+NF is on the left, Mโˆ’ NF on the right. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Table 3.8: The normal form transformation A up to order three. The component A+ is on the left, Aโˆ’ on the right. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Table 3.9: Tune and coefficients of amplitude โˆš and parameter ๐›ฟ๐œ” dependent tune shifts for centrifugal governor with ๐œ”0 = 2. . . . . . . . . . . . . . . . . . . . . . . . . 58 Table 4.1: The expansion of H๐‘ง (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ = 0) and ๐ธ (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ = 0) for relative bounded motion orbits with an average nodal period ๐‘‡๐‘‘ = 7.64916169 (โ‰ˆ103 min) and an average ascending node drift of ฮ”ฮฉ = 1.22871195E-3 rad. The expansion is relative to the pseudo-circular LEO from [42]. . . . . . . . . . . . . . . . . . 78 x Table 4.2: The LEOs below are all initiated at ๐‘ฃ ๐‘Ÿ,0 = โˆ’1.05621369E-3 and ๐‘Ÿ 0 = 1.14016749 + ๐›ฟ๐‘Ÿ, and have an average nodal period of ๐‘‡๐‘‘ = 7.64916169 (โ‰ˆ103 min) and an average ascending node drift of ฮ”ฮฉ = 1.22871195E-3 rad. The pseudo-circular LEO from [42] is denoted by O0 . . . . . . . . . . . . . . . 79 Table 4.3: Expansion of ๐œ” ๐‘ (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ = 0) of relative bounded motion LEOs with an average nodal period ๐‘‡๐‘‘ = 7.64916169 (โ‰ˆ103 min) and an average node drift of ฮ”ฮฉ = 1.22871195E-3 rad. The expansion is relative to the pseudo-circular LEO from [42]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 Table 4.4: The expansion of H๐‘ง (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ = 0) and ๐ธ (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ = 0) for relative bounded motion MEOs with an average nodal period of ๐‘‡๐‘‘ = 53.5395648 (โ‰ˆ 12 h) and an average ascending node drift of ฮ”ฮฉ = โˆ’3.35410945E-4 rad. The expansion is relative to the pseudo-circular MEO from [10]. . . . . . . . . . . . . . . . . . 83 Table 4.5: The MEOs below are all initiated at ๐‘ฃ ๐‘Ÿ,0 = โˆ’1.14150072E-4 and ๐‘Ÿ 0 = 4.17198963 + ๐›ฟ๐‘Ÿ, and have an average nodal period of ๐‘‡ ๐‘‘ = 53.5395648 (โ‰ˆ 12 h) and an average ascending node drift of ฮ”ฮฉ = โˆ’3.35410945E-4 rad. The orbit O0 is the pseudo-circular MEO from [10]. . . . . . . . . . . . . . . . 83 Table 4.6: Expansion of ๐œ” ๐‘ (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ = 0) of relative bounded motion orbits with an average nodal period of ๐‘‡ ๐‘‘ = 53.5395648 (โ‰ˆ 12 h) and an average ascending node drift of ฮ”ฮฉ = โˆ’3.35410945E-4 rad. The expansion is relative to the pseudo-circular MEO from [10]. . . . . . . . . . . . . . . . . . . . . . . . . . . 84 Table 4.7: The following orbit parameters are obtained by evaluating H๐‘ง (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ = 0) and ๐ธ (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ = 0) from Tab. 4.1 and Tab. 4.4 for various ๐›ฟ๐‘Ÿ keeping ๐›ฟ๐‘ฃ ๐‘Ÿ = 0. . . . . 86 Table 5.1: Percentages of different characterization groups. Read as follows: ๐‘ฅ % of Base particles have the property Property. All particles that hit a collimator during the 4500 turns of tracking are considered lost. . . . . . . . . . . . . . . . . . . . 140 Table 6.1: Verified global optimization results for ๐‘ˆโ˜… ๐‘˜,1D . The ๐‘˜ particles form ๐‘›pairs pairwise interactions. The upper bound ๐‘ˆ ๐‘˜,1D,UB on the minimum energy (see Sec. 6.2.2.3) was used to calculate ๐‘Ÿ ๐‘˜1D,UB , which sets the lower bound of the initial search domain (see Sec. 6.2.2.4 and Eq. (6.45)). The optimizer COSY-GO with QFB/LDB enabled was operated with Taylor Models of third order. The number of remaining boxes with all side-lengths ๐‘  < ๐‘ min is denoted by ๐‘›fin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 Table 6.2: Verified global optimization results for configurations of ๐‘˜ particles in 1D. The variable ๐‘ฃโ˜…๐‘ฅ,๐‘– is the optimal distance between two adjacent particles ๐‘๐‘– and ๐‘๐‘–+1 , with 1 โ‰ค ๐‘– < ๐‘˜, and the mirror symmetry can be observed. . . . . . . . . . . . . 166 xi Table 6.3: Performance of verified global optimization using COSY-GO with QFB/LDB enabled on minimum energy search of a 1D configuration of ๐‘˜ particles. The Taylor Model orders are denoted by โ€˜Oโ€™. Since QFB requires a minimum order of two, order one calculations are not listed. . . . . . . . . . . . . . . . . . . . . 168 Table 6.4: Verified global optimization results on the minimum energy ๐‘ˆโ˜… ๐‘˜,1D of symmetric configurations. The upper bound ๐‘ˆ ๐‘˜,1D,UB on the minimum energy (see Sec. 6.2.2.3) was used to calculate ๐‘Ÿ ๐‘˜1D,UB , which sets the lower bound of the initial search domain (see Sec. 6.2.2.4 and Eq. (6.45)). The optimizer COSY-GO with QFB/LDB enabled was operated with Taylor Models of third order. The number of remaining boxes with all side-lengths ๐‘  < ๐‘ min is denoted by ๐‘›fin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 Table 6.5: Verified global optimization results for symmetric configurations of ๐‘˜ particles in 1D for ๐‘˜ = 3 to ๐‘˜ = 20. ๐‘ฃโ˜… ๐‘ฅ,๐‘– is the optimal distance between two adjacent particles ๐‘๐‘– and ๐‘๐‘–+1 , and ๐‘ ๐‘˜โˆ’๐‘–โˆ’1 and ๐‘ ๐‘˜โˆ’๐‘– . The results for ๐‘˜ = 21 to ๐‘˜ = 26 are listed in Tab. 6.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 Table 6.6: Verified global optimization results for symmetric configurations of ๐‘˜ particles in 1D for ๐‘˜ = 21 to ๐‘˜ = 26. ๐‘ฃโ˜… ๐‘ฅ,๐‘– is the optimal distance between two adjacent particles ๐‘๐‘– and ๐‘๐‘–+1 , and ๐‘ ๐‘˜โˆ’๐‘–โˆ’1 and ๐‘ ๐‘˜โˆ’๐‘– . The results for ๐‘˜ = 3 to ๐‘˜ = 20 are listed in Tab. 6.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 Table 6.7: Performance of verified global optimization using COSY-GO with QFB/LDB enabled for the minimum energy search of a 1D symmetric configuration of ๐‘˜ particles. The Taylor Model orders are denoted by โ€˜Oโ€™. Note that we use ๐‘›var as a shorthand notation for ๐‘›1D,sym,var in the table. . . . . . . . . . . . . . . . . 174 Table 6.8: Verified global optimization results for the minimum energy configurations of four particles in 2D, S4,2D โ˜… . The ๐‘Ÿ โ˜… yield the optimal distances between ๐‘–๐‘— particles ๐‘๐‘– and ๐‘ ๐‘— . ๐‘ฃโ˜… ๐‘ฅ,๐‘– is the optimal ๐‘ฅ distance between particles ๐‘๐‘– and โ˜… ๐‘๐‘–+1 , and ๐‘ฃ ๐‘ฆ,๐‘– is the optimal ๐‘ฆ position of particle ๐‘๐‘– . . . . . . . . . . . . . . . . 190 Table 6.9: Verified global optimization results for the minimum energy configurations of five particles in 2D, S5,2D โ˜… . The ๐‘Ÿ โ˜… yield the optimal distance between particles ๐‘–๐‘— ๐‘๐‘– and ๐‘ ๐‘— . ๐‘ฃโ˜… ๐‘ฅ,๐‘– is the optimal ๐‘ฅ distance between particles ๐‘๐‘– and ๐‘๐‘–+1 and ๐‘ฃโ˜…๐‘ฆ,๐‘– is the optimal ๐‘ฆ position of particle ๐‘๐‘– . . . . . . . . . . . . . . . . . . . . . . . . 192 Table 6.10: Performance of verified global optimization using COSY-GO with QFB/LDB enabled on minimum energy search of a 2D configuration of ๐‘˜ particles. The Taylor Model orders are denoted by โ€˜Oโ€™. . . . . . . . . . . . . . . . . . . . . . . 192 xii Table 6.11: Performance of verified global optimization using parallel COSY-GO with QFB/LDB enabled for minimum energy search of a 2D configuration of six particles, S6,2D โ˜… . The parallel computations are run on Cori at NERSC using different communication timing ๐‘กcom . . . . . . . . . . . . . . . . . . . . . . . . 193 Table 6.12: Verified global optimization results for the minimum energy configurations of six particles in 2D, S6,2D โ˜… . The ๐‘Ÿ โ˜… yield the optimal distance between particles ๐‘–๐‘— ๐‘๐‘– and ๐‘ ๐‘— . ๐‘ฃโ˜… ๐‘ฅ,๐‘– is the optimal ๐‘ฅ distance between particles ๐‘๐‘– and ๐‘๐‘–+1 and ๐‘ฃโ˜…๐‘ฆ,๐‘– is the optimal ๐‘ฆ position of particle ๐‘๐‘– . . . . . . . . . . . . . . . . . . . . . . . . 195 Table 6.13: Verified global optimization results for the minimum energy configurations of seven particles in 2D, S7,2D โ˜… . The ๐‘Ÿ โ˜… yield the optimal distance between ๐‘–๐‘— particles ๐‘๐‘– and ๐‘ ๐‘— . ๐‘ฃโ˜… ๐‘ฅ,๐‘– is the optimal ๐‘ฅ distance between particles ๐‘๐‘– and ๐‘๐‘–+1 โ˜… and ๐‘ฃ ๐‘ฆ,๐‘– is the optimal ๐‘ฆ position of particle ๐‘๐‘– . The values below are for the configuration (๐œ…, ๐œˆ) = (5, 6) from Fig. 6.23. . . . . . . . . . . . . . . . . . . . . 197 Table 6.14: Verified global optimization results for the minimum energy configurations of five particles in 3D, S5,3Dโ˜… . The ๐‘Ÿ โ˜… yield the optimal distance between particles ๐‘–๐‘— ๐‘๐‘– and ๐‘ ๐‘— . ๐‘ฃ ๐‘ฅ,๐‘– is the optimal ๐‘ฅ distance between particles ๐‘๐‘– and ๐‘๐‘–+1 . ๐‘ฃโ˜…๐‘ฆ,๐‘– โ˜… and ๐‘ฃโ˜… ๐‘ง,๐‘– are the optimal ๐‘ฆ and ๐‘ง positions of particle ๐‘๐‘– . . . . . . . . . . . . . . . 200 Table 6.15: Summary of verified global optimization results on the minimum energy of configurations in 2D and 3D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 Table 6.16: Analysis of the normal form radius range of the five island particles in Fig. 5.30 and the number of turns it requires to get from the lower end of the range to the upper end. The islands are numbered from smallest (1) to largest (5). . . . . . . 218 Table 6.17: Number of turns required by the islands from Fig. 5.30 to cross the given onion layers in ๐‘Ÿ NF,2 direction. The islands are numbered from smallest (1) to largest (5). Additionally, the minimum number of turns required to cross the onion layer determined by the verified analysis is shown. . . . . . . . . . . . . . . . . 219 Table A.1: The left columns list the variables of [59], denoted by ๐‘ฃ0ยท,๐‘– , and their respective initial search domains. The middle columns list the variables of our optimization and their respective initial search domains. The right columns show the optimized variables of our optimization. . . . . . . . . . . . . . . . . . . . . . . 245 xiii LIST OF FIGURES Figure 2.1: Schematic illustration of the various normal form quantities involved in the calculation of the minimum iteration number within allowed region D. . . . . . 21 Figure 2.2: Verified representation of ๐‘“ (๐‘ฅ) = sin(๐œ‹๐‘ฅ/2) โˆ’ exp(๐‘ฅ) over the domain D = ๐ผ1 = [โˆ’1, 1] with interval methods using ๐‘“ (D) and with Taylor Models ( P๐‘š, ๐‘“ , ๐œ– ๐‘š,D, ๐‘“ ) of various orders ๐‘š. The original function ๐‘“ (๐‘ฅ) is indicated by the black line, while its DA polynomial representation is shown in green. The bounds at a distance ๐œ– ๐‘š,D, ๐‘“ from the DA polynomial are red. The two straight blue lines indicate the bounds of the interval evaluation. Note that the scale of the ๐‘ฆ axis is changing to better illustrate the tightness of the Taylor Model representation with higher orders. Accordingly, the interval bounds are only shown for order ๐‘š = 1 and order ๐‘š = 2. . . . . . . . . . . . . . . . . . . 26 Figure 3.1: Schematic illustration of centrifugal governor. . . . . . . . . . . . . . . . . . . 30 Figure 3.2: Illustration of the stable equilibrium angle ๐œ™0 of the arms of the centrifugal governor as a function of the rotation frequency ๐œ”. For ๐œ” > ๐œ”min = 1, ๐œ™0 = 0 is an unstable equilibrium angle. . . . . . . . . . . . . . . . . . . . . . . . . . 32 Figure 3.3: Potential well of ๐‘ˆeff for multiple oscillation frequencies ๐œ”. The equilibrium angle ๐œ™0 corresponds to the minimum of the potential well. . . . . . . . . . . . 34 โˆš Figure 3.4: Dynamics of the centrifugal governor for a rotation frequency of ๐œ” = 2. The centrifugal governor arms were initiated with ๐œ™ยค = ๐‘ ๐œ™ = 0 and at the following angles: 60ยฐ, 65.5ยฐ, 69.5ยฐ, 73.5ยฐ, 77.5ยฐ, 81.5ยฐ, 85.5ยฐ, and 89.5ยฐ. The left plot shows the oscillatory behavior around the equilibrium angle at ๐œ™0 = 60ยฐ over time. The right plot shows the stroboscopic phase space behavior from repetitive map evaluation. To relate the phase space behavior to the position behavior in time, the ๐œ™ axis of both plots are aligned. . . . . . . . . . . . . . . 35 Figure 3.5: Phase space behavior โˆš of the centrifugal governor arms around their equilibrium angle of ๐œ™0 (๐œ” = 2) = 60ยฐ provided by a tenth order Poincarรฉ map of the system. a) shows the original phase space behavior. b) shows the associated circular behavior in normal form. . . . . . . . . . . . . . . . . . . . . . . . . . 39 Figure 3.6: Comparison between the calculated period with normal form methods ๐‘‡NF = 1/๐œˆ(๐‘ž 0 , ๐‘ 0 ) for calculation order ten (O10) and calculation order three (O3) to the actual period of oscillationโˆšgiven by the oscillatory behavior of the centrifugal governor arms for ๐œ” = 2 from Fig. 3.4. . . . . . . . . . . . . . . . 57 xiv Figure 3.7: Phase space tracking of incomplete normal form maps of order ten โˆš of the centrifugal governor arms with a fixed rotation frequency of ๐œ” = 2. The original map (a), only linear normal form transformation (b), and only normal form transformations up to order two (c) and three (d), respectively. The normal form up to the full tenth order was illustrated in Fig. 3.5. . . . . . . . . . 60 Figure 4.1: The behavior of the Keplerian elements of a low Earth orbit under zonal gravitational perturbations up to ๐ฝ15 (purple) and as a regular Kepler orbit in the unperturbed gravitational field (green) over time. Left and right plots show different time scales of the behavior. Note that the vertical scales of ฮฉ and ๐œ” are adjusted in the right plot to show the long term behavior. . . . . . . . 68 Figure 4.2: Keplerian elements of a quasi-circular low Earth orbit under Earthโ€™s zonal gravitational perturbation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 Figure 4.3: a) Distorted phase space behavior in the original phase space (๐‘ž, ๐‘) and b) circular behavior in the corresponding normal form phase space (๐‘ž NF , ๐‘ NF ). In a), the phase space angle advancement ฮ› ๐‘˜ and the phase space radius ๐‘Ÿ๐‘– are not constant by continuously change along each of the phase space curves. In b), the phase space behavior is rotationally invariant (โ€˜normalizedโ€™) with a constant radius ๐‘Ÿ NF and a constant but amplitude dependent angle advancement ฮ›(๐‘Ÿ NF ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Figure 4.4: Oscillatory behavior of the bounded motion quantities ๐‘‡๐‘‘ and ฮ”ฮฉ of the bounded LEOs O1 and O2 initiated at ๐›ฟ๐‘Ÿ = 0.06 and ๐›ฟ๐‘Ÿ = 0.13, respectively. Additionally, the constant nodal period ๐‘‡๐‘‘โ˜… = 7.64916169 and constant ascending node drift of ฮ”ฮฉโ˜… = 0.0704โ—ฆ of the fixed point orbit O0 are shown. The periods of oscillation are 1763 orbital revolutions (126 days) for O2 , 1810 orbital revolutions (129 days) for O1 , and 1823 orbital revolutions (130 days) for ๐›ฟ๐‘Ÿ โ†’ 0 of O0 . The shown results are generated by numerical integration. . . 79 Figure 4.5: Relative bounded motion of LEOs with an average nodal period of ๐‘‡๐‘‘ = 7.64916169 (โ‰ˆ103 min) and an average node drift of ฮ”ฮฉ = 1.22871195E-3 rad for 14 years. The total relative distance between the orbits is shown in the left plot and the right plot shows the relative radial and along-track distance between orbit pairs from the perspective of one of the orbits in the pair. The oscillation in the relative distance between O2 and O1 is caused by the rotating orbital orientation of the orbits at different frequencies. . . . . . . . . . . . . . 80 xv Figure 4.6: Oscillatory behavior of the bounded motion quantities ๐‘‡๐‘‘ and ฮ”ฮฉ of the bounded MEOs O1 and O2 initiated at ๐›ฟ๐‘Ÿ = 0.24 and ๐›ฟ๐‘Ÿ = 0.52, respectively. Additionally, the constant nodal period ๐‘‡๐‘‘โ˜… = 53.5395648 and constant ascending node drift of ฮ”ฮฉโ˜… = โˆ’0.0192176316 deg of the fixed point orbit O0 are shown. The periods of oscillation are 38682 orbital revolutions (52.9 years) for O2 , 34621 orbital revolutions (47.4 years) for O1 , and 33671 orbital revolutions (46.1 years) for ๐›ฟ๐‘Ÿ โ†’ 0 of O0 . The shown results are generated by numerical integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 Figure 4.7: Relative bounded motion of MEOs from Tab. 4.5 with an average nodal period of ๐‘‡๐‘‘ = 53.5395648 (โ‰ˆ 12 h) and an average ascending node drift of ฮ”ฮฉ = โˆ’3.35410945E-4 rad over 70 years. The total relative distance between the orbits is shown in the left plots and the right plot shows the relative radial and along-track distance between orbit pairs from the perspective of one of the orbits in the pair. The โ€˜breathingโ€™ of the relative total distance between O2 and O0 originates from the rotating orbital orientation of pseudo-elliptical O2 relative to the pseudo-circular O0 . Due to the very long rotation periods, only the first 70 years of the relative distance oscillation and radial/along-track behavior between O2 and O1 could be shown. . . . . . . . . . . . . . . . . . . 85 Figure 4.8: The behavior of the bounded motion quantities ๐‘‡๐‘‘ and ฮ”ฮฉ for the test orbits from Tab. 4.7 of the calculated LEO bounded motion set generated by numerical integration. For large ๐›ฟ๐‘Ÿ, the influences of higher order oscillations are apparent. The frequency and amplitude of oscillation increase with increasing ๐›ฟ๐‘Ÿ. The amplitude of ฮ”ฮฉ is particularly sensitive to ๐›ฟ๐‘Ÿ. . . . . . . . 87 Figure 4.9: Distance between the orbits in the calculated bounded motion set and O0 is determined in regular time intervals with numerical integration over more than ten years. The left plot only shows the upper bound to avoid overlaps. Thin horizontal lines at the initial upper bound emphasize small changes. The dotted light blue curve (right) originates from an unintended near-resonance between the chosen time interval for distance evaluations and the orbital behavior. A measurable increase in relative distances (left) over 10 years for ๐›ฟ๐‘Ÿ โ‰ฅ 0.3 is supported by thickening curves in the radial/along-track behavior (right). . . . . 88 Figure 4.10: Behavior of the bounded motion quantities ๐‘‡๐‘‘ and ฮ”ฮฉ for the test orbits from Tab. 4.7 of the calculated MEO bounded motion set generated by numerical integration. In contrast to the investigated LEOs, the frequency and amplitude of oscillation decrease with increasing ๐›ฟ๐‘Ÿ such that O1.4 appears almost steady. For ๐›ฟ๐‘Ÿ โ‰ฅ 0.8 the center of oscillation of ฮ”ฮฉ start to drift to more negative values and away from ฮ”ฮฉโ˜…. To capture both, the oscillatory behavior around ฮ”ฮฉโ˜… and the drift of the center of oscillation for very large ๐›ฟ๐‘Ÿ, two plots with a different scale and range are shown for ฮ”ฮฉ. . . . . . . . . . . . . . . . . . . . 89 xvi Figure 4.11: Distance between the orbits in the calculated bounded motion set and O0 is determined in regular time intervals by numerical integration over more than 70 years. The left plot only shows the upper bound to avoid overlaps. Thin horizontal lines at the initial upper bound emphasize small changes. The โ€˜breathingโ€™ of the total relative distance from the orbital rotation is clearly visible. Its period increases with increasing ๐›ฟ๐‘Ÿ until being unrecognizable due to the strong divergence for ๐›ฟ๐‘Ÿ โ‰ฅ 1.4, which is supported by thinker curves in the right plot. The weaker divergence over the 70-year timespan is already noticeable for ๐›ฟ๐‘Ÿ โ‰ฅ 0.9. The divergence is caused by the offset in respective bounded motion quantities (see Fig. 4.10). . . . . . . . . . . . . . . . . . . . . 90 Figure 5.1: The fixed points of Poincarรฉ return maps from various azimuthal locations around the ring indicate the behavior of the closed orbit (for ๐›ฟ ๐‘ = 0). The projections of the four dimensional fixed points into subspaces illustrate the influence of the magnetic field perturbations on the closed orbit around the ring. The results from the five collimator locations (C1-C5) are highlighted with red color. The ๐‘ฅ coordinate corresponds to displacements in the radially outward direction, while the ๐‘ฆ coordinate indicates the displacement in the vertically upward direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Figure 5.2: Changes of the closed orbits due to relative changes ๐›ฟ ๐‘ in the total initial momentum. The plots illustrate absolute coordinates with respect to the ideal orbit at the center of the ring for the five collimator locations (C1-C5). . . . . . 100 Figure 5.3: Phase space behavior of four particles in different phase space regions with various amplitudes and momentum offsets. Particle 4 (yellow) hits the collimator (circle in the ๐‘ฅ๐‘ฆ plot) and is lost. The momentum dependent radial position ๐‘ฅ of the particles is particularly prominent. The  individual particles are characterized by the parameter set ๐‘ฅ amp , ๐‘ฆ amp , ๐›ฟ ๐‘ . For particle 1 (P1) the parameter set is (6 mm, 12 mm, โˆ’0.39%). For particle 2 (P2) the parameter set is (12 mm, 6 mm, โˆ’0.39%) . For particle 3 (P3) the parameter set is (27 mm, 16 mm, +0.13%). For particle 4 (P4) the parameter set is (6 mm, 25 mm, +0.39%). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Figure 5.4: Schematic illustration of viable ๐‘ฅ๐‘ฆ region around a momentum dependent fixed point. The region contains all rectangles centered at the fixed point, which do not overlap with the collimator circle. . . . . . . . . . . . . . . . . . 103 Figure 5.5: Vertical and horizontal tune dependence in the model of the Muon ๐‘”-2 Storage Ring of E989 on relative offsets ๐›ฟ ๐‘ from the reference momentum ๐‘ 0 . . . . . . 104 xvii Figure 5.6: Amplitude dependent tune shifts in the model of the Muon ๐‘”-2 Storage Ring of E989. The black line indicates the amplitude dependent tune shifts for ๐›ฟ ๐‘ = 0, while the other lines have a momentum offset specified by their color. For the left plots regarding the radial amplitude dependence, the vertical amplitude relative to the momentum dependent fixed point is set to zero and vice versa for the plots regarding the vertical amplitude dependence on the right. The lines end when the total ๐‘ฅ๐‘ฆ amplitude of the particle relative to the ideal orbit reaches the collimator at ๐‘Ÿ 0 = 45 mm. . . . . . . . . . . . . . . . . . . . . . . . 106 Figure 5.7: Behavior of combined amplitude dependent tune shifts at multiple momentum offsets for an ESQ voltage of 18.3 kV. . . . . . . . . . . . . . . . . . . . . . . 108 Figure 5.8: Behavior of combined amplitude dependent tune shifts at multiple momentum offsets for an ESQ voltage of 18.3 kV. . . . . . . . . . . . . . . . . . . . . . . 109 Figure 5.9: Behavior of combined amplitude dependent tune shifts at multiple momentum offsets for an ESQ voltage of 18.3 kV. . . . . . . . . . . . . . . . . . . . . . . 110 Figure 5.10: Projections of the distribution of the variables (๐‘ฅ, ๐‘Ž, ๐‘ฆ, ๐‘, ๐›ฟ ๐‘) in the realistic beam simulation at the azimuthal ring location of the central kicker. . . . . . . . 112 Figure 5.11: The tune footprint of a realistic beam distribution at the azimuthal ring location of the central kicker. The tune footprint from the 10th order calculation is colored according to the momentum offset of the individual particles. The black lines correspond to resonance conditions. In a) the 8th order calculation (green) is overlaid to illustrate the drastic influence of the strong ninth order nonlinearities of the map caused by the 20th-pole of the ESQ potential. In b) the particles with a momentum offset โˆ’0.3% < ๐›ฟ ๐‘ < 0.1% are overlaid in green. In c) the particles with a momentum offset 0.1% < ๐›ฟ ๐‘ < 0.28% are overlaid in green. In d) the particles with a momentum offset 0.28% < ๐›ฟ ๐‘ < 0.5% are overlaid in green. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Figure 5.12: The left plot shows in violet the ratio of particles that have a larger ๐‘‘NF,max then the corresponding ๐‘ฅ value. The green boxes indicate the ratio of particles lost in each ๐‘‘NF,max group, e.g., particles that encounter a maximum normal form defect larger than 2โˆ’7 = 7.8 ยท 10โˆ’3 are all lost (loss ratio of 1). The right plot shows the ratio between the normal form radius range over 4500 turns (๐‘‘NF,lt ) and the maximum encountered normal form defect ๐‘‘NF,max for each particle. The particles are grouped into surviving and lost particles. . . . . . . . 115 xviii Figure 5.13: The plots show the long term normal form defect dependent on the calculated tune range of each particle. The dots are the minimum calculated tune of each particle while tracking. Red dots indicate that the respective particle is lost over the 4500 tracking turns. The gray lines show the calculated tune range of each particle. The left plot illustrates the radial long term normal form defect with respect to the radial tune and the 17/18 resonance (green line). The right plot shows the vertical long normal form defect with respect to the vertical tune and the 1/3 resonance (green line). . . . . . . . . . . . . . . . . . . . . . 116 Figure 5.14: The tune range of the particles forming the spike in Fig. 5.13 are shown on the left. The right plot shows the normal form defect of the particles depends on their closeness to the 6๐œˆ๐‘ฅ + 4๐œˆ ๐‘ฆ = 7 resonance (green line). . . . . . . . . . . . 117 Figure 5.15: The radial and vertical phase space behavior indicates that this particle (๐›ฟ ๐‘ = 0.015%) oscillates at constant amplitudes around its momentum dependent reference orbit. The overall normal form radius is constant and confirms this. Accordingly, the tune footprint of the particle is a single dot. This is a trivial large amplitude loss. . . . . . . . . . . . . . . . . . . . . . . . 119 Figure 5.16: The vertical phase space behavior of this particle (๐›ฟ ๐‘ = 0.196%) has a slight triangular deformation. The overall normal form radius indicates a modulated amplitude and the spread out tune footprint starts right after the vertical 1/3-resonance line. Despite slight influence of the resonance, the rather elliptical phase space behavior makes this a trivial large amplitude loss. . . . . . 120 Figure 5.17: This particle (๐›ฟ ๐‘ = โˆ’0.088%) is caught around a period-3 fixed point structure in the vertical phase space, which is related to the vertical 1/3-resonance. We refer to these structures as islands and the loss mechanisms is called island related loss. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Figure 5.18: This particle (๐›ฟ ๐‘ = โˆ’0.015%) forms large islands around a period-3 fixed point structure in the vertical phase space, which is associated with a major modulation of the oscillation amplitude. . . . . . . . . . . . . . . . . . . . . . 122 Figure 5.19: This particle (๐›ฟ ๐‘ = โˆ’0.127%) jumps between the islands. The large radial amplitude and/or the closeness to the (17/18, 1/3) resonance point might have triggered the jump. This is an example of moderate unstable behavior around a period-3 fixed point structure. . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Figure 5.20: This particle (๐›ฟ ๐‘ = 0.024%) shows a different kind of moderate unstable behavior around a period-3 fixed point structure, where the island size varies. The particle has both, a large radial amplitude and the closeness to the (17/18, 1/3) resonance point. . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 xix Figure 5.21: This particle (๐›ฟ ๐‘ = 0.140%) forms a shuriken like shape in the vertical phase space. In this pattern there are two period-3 fixed point structures involved indicated by the double crossing of the vertical 1/3 resonance line. . . . . . . . 125 Figure 5.22: This particle (๐›ฟ ๐‘ = 0.196%) illustrates moderate unstable behavior in a shuriken pattern. The radial amplitude is not particularly large, but the resonance point (17/18, 1/3) is very close, which might be the trigger of the unsuitability. The unstable behavior is also visible in the continuously increasing normal form radius. . . . . . . . . . . . . . . . . . . . . . . . . . . 126 Figure 5.23: This particle (๐›ฟ ๐‘ = 0.242%) illustrates a shuriken pattern, where the two period-3 fixed point structures are more obvious. The muon experiences a major modulation in the vertical oscillation amplitude and performs a double crossing of the vertical 1/3 resonance line. . . . . . . . . . . . . . . . . . . . . 127 Figure 5.24: This particle (๐›ฟ ๐‘ = โˆ’0.096%) shows a shuriken pattern with unstable ten- dencies. The large radial amplitude and/or the closeness to the radial 17/18 resonance line might be the trigger for the instability. . . . . . . . . . . . . . . 128 Figure 5.25: This particle (๐›ฟ ๐‘ = โˆ’0.159%) shows a shuriken pattern with a moderate instability. The two period-3 fixed point structures are so close together that the particle gets temporarily caught around the inner one of them in an island pattern. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Figure 5.26: This particle (๐›ฟ ๐‘ = 0.181%) shows the pattern of a very blunt shuriken. The vertical amplitude oscillation is only moderate and illustrates there can be almost regular behavior between two period-3 fixed point structures. . . . . . . 130 Figure 5.27: This particle (๐›ฟ ๐‘ = 0.106%) is characterized by a very large vertical amplitude, which is additionally modulated by the shuriken pattern. Its one of the very few particles for which the orbit considerably overlaps with the collimator boundary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Figure 5.28: This particle (๐›ฟ ๐‘ = 0.118%) shows strong instabilities caused by a combination of a very large vertical amplitude in combination with a period-3 fixed point structure, which occasionally captures the orbit in an island pattern. . . . . . . . 132 Figure 5.29: This particle (๐›ฟ ๐‘ = 0.010%) diverges due to its unstable orbit. The approach of the unstable fixed point with such a large vertical amplitude are likely the trigger of the divergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 xx Figure 5.30: The left plot shows stroboscopic tracking in the vertical phase space illustrating orbit behavior with a single period-3 fixed point structure present. The orbits only differ in their vertical phase space behavior โ€“ they all have the same momentum offset of ๐›ฟ ๐‘ = 0.126 % and are at the momentum dependent equilibrium point in radial phase space (๐‘ฅ = 10.64 mm, ๐‘Ž = 0.045 mrad) having no radial oscillation amplitude. The blue orbits indicate the island patterns around the stable fixed points in the middle of the islands. The red orbits are right at the edge before being caught around the fixed points. The three unstable fixed points are in the space between the two red orbits, where the islands almost touch. In the right plot, the attractive (green) and repulsive (violet) eigenvectors of the linear dynamics around the unstable fixed points are schematically shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 Figure 5.31: Stroboscopic tracking in the vertical phase space illustrating orbit behavior with two period-3 fixed point structures present. The orbits in each plot only differ in their vertical phase space behavior. All orbits have the same momentum offset of ๐›ฟ ๐‘ = 0.339 %. The four plots differ by their radial amplitude around the momentum dependent equilibrium point in radial phase space at (๐‘ฅ = 27.7 mm, ๐‘Ž = 0.144 mrad). The radial amplitudes are: a) ๐‘ฅ amp = 6 mm, b) ๐‘ฅ amp = 4.8 mm, c) ๐‘ฅ amp = 4 mm, d) ๐‘ฅamp = 1 mm. The blue orbits indicate the island patterns around the stable fixed points. The red orbits are right at the edge before being caught around the period-3 fixed points. The green orbits are caught around both period-3 fixed point structures. The gray orbits in d) emphasize that half of the fixed points from c) have indeed been annihilated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Figure 5.32: a) Shows how the muon loss ratio is composed of regular particles (purple) and particles involved with period-3 fixed point structures (green). Of the period-3 particles (green), the fraction caught in islands structures is indicated by the blue stripe pattern. In b) the loss ratio over time is shown for each subgroup of lost particles to better understand which losses drive to overall loss from plot a). The tracking starts after the initial 30 ๐œ‡s of final beam preparation when data taking is initiated. . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 Figure 6.1: A contour plot of the Rosenbrock function with (๐‘Ž, ๐‘) = (1, 100). . . . . . . . . 144 Figure 6.2: Projections of the multidimensional generalizations of the Rosenbrock function (Eq. (6.2)) into 2D-subspaces around minimum at ๐‘ฅยฎ = (1, 1, ..., 1), i.e., all variables are equal one except for the ones shown in the respective plot. . . . . . 146 Figure 6.3: Verified global optimization of the 2D Rosenbrock function using COSY-GO in different operation modes with fourth order Taylor Models for all modes except interval evaluations (IN). . . . . . . . . . . . . . . . . . . . . . . . . . . 148 xxi Figure 6.4: No cluster effect for the COSY-GO operating mode QFB/LDB, but a significant cluster effect for the IN evaluation. . . . . . . . . . . . . . . . . . . . . . . . . 149 Figure 6.5: Splitting comparison between fourth order Taylor Model approach with QFB/LDB enabled and interval evaluation using the example of the modified 2D Rosenbrock function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 Figure 6.6: Time consumption and number of steps in the optimization of the regular ๐‘› dimensional Rosenbrock function from Eq. (6.2) at various orders with COSY-GO and QFB/LDB enabled. Additionally, the interval evaluation performance is also shown for comparison. . . . . . . . . . . . . . . . . . . . . 151 Figure 6.7: Time consumption and number of steps in the optimization of the ๐‘› dimensional Rosenbrock function with an additional artificial dependency problem ๐‘“ = ๐‘“2D โˆ’ ๐‘“2D + ๐‘“2D for various Taylor Model orders with COSY-GO and QFB/LDB enabled. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 Figure 6.8: The Lennard-Jones potential for a pairwise interaction between two particles as defined in Eq. (6.8). The potential well around the minimum is shown on the left, while the right plot emphasizes its shallowness compared to the steep potential wall for ๐‘Ÿ < 1. The potential is offset for the convenience of calculation so that the single minimum ๐‘ˆLJ โ˜… has an energy of zero. . . . . . . . . 154 Figure 6.9: The particles ๐‘๐‘– are numbered according to their ๐‘ฅ position. The variable ๐‘ฃ ๐‘ฅ,๐‘™ denotes the distance between particle ๐‘ ๐‘™ and ๐‘ ๐‘™+1 . A configuration with any ๐‘ฃ ๐‘ฅ,๐‘™ > 1 (left picture) is never optimal, because the ๐‘ˆ ๐‘˜ can always be lowered by setting ๐‘ฃ ๐‘ฅ,๐‘™ = 1 (right picture). . . . . . . . . . . . . . . . . . . . . . . . . . 159 Figure 6.10: The relation between ๐‘ˆLJ and the corresponding inter-particle distance(s). Note that ๐‘Ÿ max (๐‘ˆLJ ) is only defined for ๐‘ˆLJ โ‰ค 1. ๐‘Ÿ min (๐‘ˆLJ ) is only decreasing very slowly with increasing ๐‘ˆLJ as the logarithmic plot on the right shows. . . 160 Figure 6.11: The plots show the values of the optimization variables ๐‘ฃโ˜… ๐‘ฅ,๐‘– of the minimum energy configuration of ๐‘˜ particles in 1D that resulted from the verified global optimization using COSY-GO. The minimum energy configuration is mirror symmetric with the middlemost distances between adjacent particles asymptotically approaching ๐‘Ÿโ˜… โ‰ˆ 0.998724135, the solution of the infinite equidistant configuration from Eq. (6.41). The right plot shows the logarithm of the difference between the calculated distances from the verified optimization and ๐‘Ÿโ˜…. The ranges reflect the side-length of the remaining box. . . . . . . . . . 167 xxii Figure 6.12: The plots show the values for the optimized variables ๐‘ฃโ˜… ๐‘ฅ,๐‘– of the symmetric minimum energy configuration of ๐‘˜ particles that resulted from the verified global optimization. Again, the middlemost distances asymptotically approach ๐‘Ÿโ˜… โ‰ˆ 0.998724135, the solution of the infinite equidistant configuration from Eq. (6.41). The right plot shows the logarithm of the difference between the calculated distances from the verified optimization and ๐‘Ÿโ˜…. The ranges reflect the side-length of the remaining box. . . . . . . . . . . . . . . . . . . . . . . . 173 Figure 6.13: Performance of the minimum energy search for configurations of ๐‘˜ particles in 1D using COSY-GO with different Taylor Model orders with QFB/LDB enabled. The order of the Taylor Models is denoted by โ€˜Oโ€™. The results from both Sec. 6.2.2.6 and Sec. 6.2.2.7 (โ€™symโ€™) are shown. . . . . . . . . . . . . . . . 175 Figure 6.14: The particles ๐‘๐‘– are numbered according to their ๐‘ฅ position. The variable ๐‘ฃ ๐‘ฅ,๐‘™ denotes the ๐‘ฅ distance between particle ๐‘ ๐‘™ and ๐‘ ๐‘™+1 . A configuration with any ๐‘ฃ ๐‘ฅ,๐‘™ > 1 (left picture) is never optimal, because the overall potential can always be lowered by setting ๐‘ฃ ๐‘ฅ,๐‘™ = 1 (right picture). . . . . . . . . . . . . . . . 180 Figure 6.15: The optimal 2D configuration of five particles is denoted by five dots. Enclose all the five particles by a 2D rectangle using the minimum and maximum coordinates in ๐‘ฅ and ๐‘ฆ, shown by a solid line rectangle. Surround the resulting rectangle with a band of width 1, and we have a initial search domain for the sixth particle (shaded area). . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 Figure 6.16: Schematic illustration of the upper bound on distance perpendicular to the ๐‘ฅ axis (major axis) due to the requirement of having the longest distance between ๐‘ 1 and ๐‘ ๐‘˜ . ๐‘ 1 at ๐‘ฅ 1 = 0 and ๐‘ ๐‘˜ at ๐‘ฅ ๐‘˜ , where the major axis length ๐‘Ÿ 1๐‘˜ = ๐‘ฅ ๐‘˜ . . . 182 Figure 6.17: Initial search domain for a configuration of ๐‘˜ particles in 2D. Note that the initial domain width in ๐‘ฅ direction is always 1 (see Eq. (6.72)) and that the ๐‘ฅ position of particle ๐‘๐‘– determines the starting position in ๐‘ฅ of the domain of particle ๐‘๐‘–+1 . Particle ๐‘ 1 is fixed to the origin, particle ๐‘ 2 is bound by ๐‘ฆ 2 โ‰ค 0, and particle ๐‘ ๐‘˜ has a fixed ๐‘ฆ value of zero. . . . . . . . . . . . . . . . . . . . . 183 Figure 6.18: Piecewise defined modified Lennard-Jones potential ๐‘ˆหœ LJ,sqr shown by the black curve. The red curves shows the Lennard-Jones potential and the green line shows the tangent of this Lennard-Jones potential of ๐‘Ÿ sqr . The plot shown here is an example case with ๐‘Ÿ LB 2 = 0.92 . . . . . . . . . . . . . . . . . . . . . . 185 Figure 6.19: Taylor Model description of piecewise defined function. Each Taylor Model is represented by three lines as previously done in Fig. 2.2. The central curve denotes the polynomial part of the Taylor Model, while the curves above and below it indicate the bounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 xxiii Figure 6.20: Minimum energy configurations of four particles in 2D, S4,2D โ˜… . Interestingly, the minimum energy configuration is not a square, an obvious symmetric object with fourfold symmetry, but a rhombus with two almost equilateral triangles very slightly squished in the horizontal direction. . . . . . . . . . . . . 190 Figure 6.21: Minimum energy configuration of five particles in 2D, S5,2D โ˜… . . . . . . . . . . . 191 Figure 6.22: Minimum energy configuration of six particles in 2D, S6,2D โ˜… . . . . . . . . . . . 194 Figure 6.23: Minimum energy configuration of seven particles in 2D, S7,2D โ˜… . The configu- ration is represented by two equivalent numbering schemes. . . . . . . . . . . . 196 Figure 6.24: Minimum energy configuration of five particles in 3D, S5,3D โ˜… . The configura- tion is shown in 2D projections, the ๐‘ฅ๐‘ฆ plane projection (left), the ๐‘ฅ๐‘ง plane projection (middle), and the ๐‘ฆ๐‘ง plane projection (right). The solution consists of a central equilateral triangle spanned by the particles ๐‘ 2 , ๐‘ 3 , and ๐‘ 4 in the ๐‘ฆ๐‘ง plane, and one particle each centered above and below that triangle. In other words, it is similar to a double tetrahedron, which is slightly squished along the major axis (the ๐‘ฅ axis) increasing the side-length of the equilateral triangle in the middle to values slightly larger than one. The inter-particle distances are shown in Tab. 6.14. . . . . . . . . . . . . . . . . . . . . . . . . . 200 Figure 6.25: The left and the middle plot show the representation of an onion layer (black region) in regular phase space coordinates. The thickness of the onion layer is determined by the range in ๐‘Ÿ NF,1 and ๐‘Ÿ NF,2 as well as the range in ๐›ฟ ๐‘. For this particular example, we set ๐›ฟ ๐‘ to a fixed value of ๐›ฟ ๐‘ = 0% instead of a range. The range in the normal form radii is given by ๐‘Ÿ NF,1 โˆˆ [0.15, 0.25] and ๐‘Ÿ NF,2 โˆˆ [0.7, 0.75]. Note that the thickness in ๐‘Ÿ NF,1 is twice the thickness in ๐‘Ÿ NF,2 . Accordingly, the projection of the onion layer into the radial phase space (๐‘ฅ, ๐‘Ž) appears roughly twice as thick as the projection into the vertical phase space (๐‘ฆ, ๐‘). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 Figure 6.26: Normal form defect landscape of the radial phase space in ๐œ™NF,1 and ๐œ™NF,2 for fixed normal form amplitudes of ๐‘Ÿ NF,1 = 0.4 and ๐‘Ÿ NF,2 = 0.4, and with ๐›ฟ ๐‘ = 0%. The underlying map considers an ESQ voltage of 18.3 kV. . . . . . . 209 Figure 6.27: The normal form defect landscape of the radial (left side) and vertical (right side) phase space for multiple onion layers of zero thickness, which are char- acterized by (๐‘Ÿ NF,1 , ๐‘Ÿ NF,2 , ๐›ฟ ๐‘). The top row corresponds to (0.1, 0.2, 0.24%), the middle row corresponds to (0.2, 0.05, 0.24%), and the bottom row corre- sponds to (0.56, 0.72, 0.04%). The underlying map considers an ESQ voltage of 18.3 kV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 xxiv Figure 6.28: The normal form defect landscape of the radial (left side) and vertical (right side) phase space for multiple onion layers of zero thickness, which are char- acterized by (๐‘Ÿ NF,1 , ๐‘Ÿ NF,2 , ๐›ฟ ๐‘). The top row corresponds to (0.1, 0.2, 0.24%), the middle row corresponds to (0.2, 0.05, 0.24%), and the bottom row corre- sponds to (0.56, 0.72, 0.04%). The underlying map considers an ESQ voltage of 17.5 kV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Figure 6.29: Nonverified normal form defect for the phase space storage regions of the Muon ๐‘”-2 Storage Ring simulation with an ESQ voltage of 17.5 kV. The individual plots show different momentum ranges, clarified by the label at the top of each graph. The color scheme corresponds to the normal form defect of the specific onion layer. The white boxes for lower normal form radii indicate a normal form defect below 10โˆ’5 . The yellow boxes denote normal form defects up to 10โˆ’4 . The orange boxes correspond to normal form defects up to 10โˆ’3 . The red boxes denote normal form defects up to 10โˆ’2.5 , and the black boxes indicate normal form defects larger than that. Each onion layer corresponds to a 0.04 ร— 0.04 box in normal form space with a thickness of 0.04% in ๐›ฟ ๐‘. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 Figure 6.30: Verified normal form defect for the phase space storage regions of the Muon ๐‘”-2 Storage Ring simulation with an ESQ voltage of 17.5 kV. The individual plots show different momentum ranges, clarified by the label at the top of each graph. The color scheme corresponds to the normal form defect of the specific onion layer. The white boxes for lower normal form radii indicate a normal form defect below 10โˆ’5 . The yellow boxes denote normal form defects up to 10โˆ’4 . The orange boxes correspond to normal form defects up to 10โˆ’3 . The red boxes denote normal form defects up to 10โˆ’2.5 , and the black boxes indicate normal form defects larger than that. Each onion layer corresponds to a 0.04 ร— 0.04 box in normal form space with a thickness of 0.04% in ๐›ฟ ๐‘. . . . . 215 Figure 6.31: Verified normal form defect for the phase space storage regions of the Muon ๐‘”-2 Storage ring simulation with an ESQ voltage of 18.3 kV. The individual plots show different momentum ranges, clarified by the label at the top of each graph. The color scheme corresponds to the normal form defect of the specific onion layer. The white boxes for lower normal form radii indicate a normal form defect below 10โˆ’5 . The yellow boxes denote normal form defects up to 10โˆ’4 . The orange boxes correspond to normal form defects up to 10โˆ’3 . The red boxes denote normal form defects up to 10โˆ’2.5 , and the black boxes indicate normal form defects larger than that. Each onion layer corresponds to a 0.04 ร— 0.04 box in normal form space with a thickness of 0.04% in ๐›ฟ ๐‘. . . . . 216 xxv Figure 6.32: Nonverified normal form defect for the phase space storage regions of the Muon ๐‘”-2 Storage Ring simulation with an ESQ voltage of 18.3 kV. The individual plots show different momentum ranges which are clarified by the label at the top of each graph. The color scheme corresponds to the normal form defect of the specific onion layer. The white boxes for lower normal form radii indicate a normal form defect below 10โˆ’5 . The yellow boxes denote normal form defects up to 10โˆ’4 . The orange boxes correspond to normal form defects up to 10โˆ’3 . The red boxes denote normal form defects up to 10โˆ’2.5 , and the black boxes indicate normal form defects larger than that. Each onion layer corresponds to a 0.04 ร— 0.04 box in normal form space with a thickness of 0.04% in ๐›ฟ ๐‘. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 Figure 6.33: Behavior of combined amplitude dependent tune shifts at multiple momentum offsets for an ESQ voltage of 17.5 kV. . . . . . . . . . . . . . . . . . . . . . . 221 Figure 6.34: Behavior of combined amplitude dependent tune shifts at multiple momentum offsets for an ESQ voltage of 17.5 kV. . . . . . . . . . . . . . . . . . . . . . . 222 Figure 6.35: Behavior of combined amplitude dependent tune shifts at multiple momentum offsets for an ESQ voltage of 17.5 kV. . . . . . . . . . . . . . . . . . . . . . . 223 Figure 6.36: Difference between verified normal form defect analysis and nonverified normal form defect analysis for the phase space storage regions of the Muon ๐‘”-2 Storage Ring simulation with an ESQ voltage of 17.5 kV. The individual plots show different momentum ranges which are clarified by the label at the top of each graph. The color scheme corresponds to the difference of the evaluated normal form defects of the specific onion layer. The white boxes for lower normal form radii indicate differences below 10โˆ’5 . The yellow boxes denote differences up to 10โˆ’4 , the orange boxes correspond to differences up to 10โˆ’3 , the red boxes denote differences up to 10โˆ’2.5 and the black boxes indicate differences larger than that. Each onion layer corresponds to a 0.04 ร— 0.04 box in normal form space with a thickness of 0.04% in ๐›ฟ ๐‘. . . . 225 Figure 6.37: Difference between verified normal form defect analysis and nonverified normal form defect analysis for the phase space storage regions of the Muon ๐‘”-2 Storage Ring simulation with an ESQ voltage of 18.3 kV. The individual plots show different momentum ranges, clarified by the label at the top of each graph. The color scheme corresponds to the difference of the evaluated normal form defects of the specific onion layer. The white boxes for lower normal form radii indicate a difference below 10โˆ’5 . The yellow boxes denote differences up to 10โˆ’4 . The orange boxes correspond to differences up to 10โˆ’3 . The red boxes denote differences up to 10โˆ’2.5 , and the black boxes indicate differences larger than that. Each onion layer corresponds to a 0.04 ร— 0.04 box in normal form space with a thickness of 0.04% in ๐›ฟ ๐‘. . . . . . . . . . . . 226 xxvi Figure 6.38: Difference between the rigorously guaranteed upper bound and the lower bound of the maximum normal form defect using Taylor Model based verified global optimization. The analysis is for the phase space storage regions of the Muon ๐‘”-2 Storage Ring simulation with an ESQ voltage of 17.5 kV. The individual plots show different momentum ranges which are clarified by the label at the top of each graph. The color scheme corresponds to the difference between the upper bound and the lower bound of the maximum normal form defect of the specific onion layer. All boxes are white because the difference is below 10โˆ’5 . Each onion layer corresponds to a 0.04 ร— 0.04 box in normal form space with a thickness of 0.04% in ๐›ฟ ๐‘. . . . . . . . . . . . . . . . . . . . 227 Figure 6.39: Difference between the rigorously guaranteed upper bound and the lower bound of the maximum normal form defect using Taylor Model based verified global optimization. The analysis is for the phase space storage regions of the Muon ๐‘”-2 Storage Ring simulation with an ESQ voltage of 18.3 kV. The individual plots show different momentum ranges which are clarified by the label at the top of each graph. The color scheme corresponds to the difference between the upper bound and the lower bound of the maximum normal form defect of the specific onion layer. All boxes are white because the difference is below 10โˆ’5 . Each onion layer corresponds to a 0.04 ร— 0.04 box in normal form space with a thickness of 0.04% in ๐›ฟ ๐‘. . . . . . . . . . . . . . . . . . . . 228 Figure 6.40: Nonverified normal form defect for the phase space storage regions of the Muon ๐‘”-2 Storage Ring simulation with an ESQ voltage of 18.3 kV using the normal form transformation up to order 1 instead of the full tenth order. The individual plots show different momentum ranges, clarified by the label at the top of each graph. The color scheme corresponds to the normal form defect of the specific onion layer. The white boxes for lower normal form radii indicate a normal form defect below 10โˆ’5 . The yellow boxes denote normal form defects up to 10โˆ’4 . The orange boxes correspond to normal form defects up to 10โˆ’3 . The red boxes denote normal form defects up to 10โˆ’2.5 , and the black boxes indicate normal form defects larger than that. Each onion layer corresponds to a 0.04 ร— 0.04 box in normal form space with a thickness of 0.04% in ๐›ฟ ๐‘. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 xxvii Figure 6.41: Nonverified normal form defect for the phase space storage regions of the Muon ๐‘”-2 Storage Ring simulation with an ESQ voltage of 18.3 kV using the normal form transformation up to order 2 instead of the full tenth order. The individual plots show different momentum ranges, clarified by the label at the top of each graph. The color scheme corresponds to the normal form defect of the specific onion layer. The white boxes for lower normal form radii indicate a normal form defect below 10โˆ’5 . The yellow boxes denote normal form defects up to 10โˆ’4 . The orange boxes correspond to normal form defects up to 10โˆ’3 . The red boxes denote normal form defects up to 10โˆ’2.5 , and the black boxes indicate normal form defects larger than that. Each onion layer corresponds to a 0.04 ร— 0.04 box in normal form space with a thickness of 0.04% in ๐›ฟ ๐‘. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 Figure 6.42: Nonverified normal form defect for the phase space storage regions of the Muon ๐‘”-2 Storage Ring simulation with an ESQ voltage of 18.3 kV using the normal form transformation up to order 3 instead of the full tenth order. The individual plots show different momentum ranges, clarified by the label at the top of each graph. The color scheme corresponds to the normal form defect of the specific onion layer. The white boxes for lower normal form radii indicate a normal form defect below 10โˆ’5 . The yellow boxes denote normal form defects up to 10โˆ’4 . The orange boxes correspond to normal form defects up to 10โˆ’3 . The red boxes denote normal form defects up to 10โˆ’2.5 , and the black boxes indicate normal form defects larger than that. Each onion layer corresponds to a 0.04 ร— 0.04 box in normal form space with a thickness of 0.04% in ๐›ฟ ๐‘. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 Figure 6.43: Nonverified normal form defect for the phase space storage regions of the Muon ๐‘”-2 Storage Ring simulation with an ESQ voltage of 18.3 kV using the normal form transformation up to order 4 instead of the full tenth order. The individual plots show different momentum ranges, clarified by the label at the top of each graph. The color scheme corresponds to the normal form defect of the specific onion layer. The white boxes for lower normal form radii indicate a normal form defect below 10โˆ’5 . The yellow boxes denote normal form defects up to 10โˆ’4 . The orange boxes correspond to normal form defects up to 10โˆ’3 . The red boxes denote normal form defects up to 10โˆ’2.5 , and the black boxes indicate normal form defects larger than that. Each onion layer corresponds to a 0.04 ร— 0.04 box in normal form space with a thickness of 0.04% in ๐›ฟ ๐‘. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 xxviii Figure 6.44: Nonverified normal form defect for the phase space storage regions of the Muon ๐‘”-2 Storage Ring simulation with an ESQ voltage of 18.3 kV using the normal form transformation up to order 5 instead of the full tenth order. The individual plots show different momentum ranges, clarified by the label at the top of each graph. The color scheme corresponds to the normal form defect of the specific onion layer. The white boxes for lower normal form radii indicate a normal form defect below 10โˆ’5 . The yellow boxes denote normal form defects up to 10โˆ’4 . The orange boxes correspond to normal form defects up to 10โˆ’3 . The red boxes denote normal form defects up to 10โˆ’2.5 , and the black boxes indicate normal form defects larger than that. Each onion layer corresponds to a 0.04 ร— 0.04 box in normal form space with a thickness of 0.04% in ๐›ฟ ๐‘. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 Figure 6.45: Nonverified normal form defect for the phase space storage regions of the Muon ๐‘”-2 Storage Ring simulation with an ESQ voltage of 18.3 kV using the normal form transformation up to order 6 instead of the full tenth order. The individual plots show different momentum ranges, clarified by the label at the top of each graph. The color scheme corresponds to the normal form defect of the specific onion layer. The white boxes for lower normal form radii indicate a normal form defect below 10โˆ’5 . The yellow boxes denote normal form defects up to 10โˆ’4 . The orange boxes correspond to normal form defects up to 10โˆ’3 . The red boxes denote normal form defects up to 10โˆ’2.5 , and the black boxes indicate normal form defects larger than that. Each onion layer corresponds to a 0.04 ร— 0.04 box in normal form space with a thickness of 0.04% in ๐›ฟ ๐‘. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 Figure 6.46: Nonverified normal form defect for the phase space storage regions of the Muon ๐‘”-2 Storage Ring simulation with an ESQ voltage of 18.3 kV using the normal form transformation up to order 7 instead of the full tenth order. The individual plots show different momentum ranges, clarified by the label at the top of each graph. The color scheme corresponds to the normal form defect of the specific onion layer. The white boxes for lower normal form radii indicate a normal form defect below 10โˆ’5 . The yellow boxes denote normal form defects up to 10โˆ’4 . The orange boxes correspond to normal form defects up to 10โˆ’3 . The red boxes denote normal form defects up to 10โˆ’2.5 , and the black boxes indicate normal form defects larger than that. Each onion layer corresponds to a 0.04 ร— 0.04 box in normal form space with a thickness of 0.04% in ๐›ฟ ๐‘. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 xxix Figure 6.47: Nonverified normal form defect for the phase space storage regions of the Muon ๐‘”-2 Storage Ring simulation with an ESQ voltage of 18.3 kV using the normal form transformation up to order 8 instead of the full tenth order. The individual plots show different momentum ranges, clarified by the label at the top of each graph. The color scheme corresponds to the normal form defect of the specific onion layer. The white boxes for lower normal form radii indicate a normal form defect below 10โˆ’5 . The yellow boxes denote normal form defects up to 10โˆ’4 . The orange boxes correspond to normal form defects up to 10โˆ’3 . The red boxes denote normal form defects up to 10โˆ’2.5 , and the black boxes indicate normal form defects larger than that. Each onion layer corresponds to a 0.04 ร— 0.04 box in normal form space with a thickness of 0.04% in ๐›ฟ ๐‘. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 Figure 6.48: Nonverified normal form defect for the phase space storage regions of the Muon ๐‘”-2 Storage Ring simulation with an ESQ voltage of 18.3 kV using the normal form transformation up to order 9 instead of the full tenth order. The individual plots show different momentum ranges, clarified by the label at the top of each graph. The color scheme corresponds to the normal form defect of the specific onion layer. The white boxes for lower normal form radii indicate a normal form defect below 10โˆ’5 . The yellow boxes denote normal form defects up to 10โˆ’4 . The orange boxes correspond to normal form defects up to 10โˆ’3 . The red boxes denote normal form defects up to 10โˆ’2.5 , and the black boxes indicate normal form defects larger than that. Each onion layer corresponds to a 0.04 ร— 0.04 box in normal form space with a thickness of 0.04% in ๐›ฟ ๐‘. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 Figure 6.49: Nonverified normal form defect for the phase space storage regions of the Muon ๐‘”-2 Storage Ring simulation with an ESQ voltage of 18.3 kV using the normal form transformation up to the full tenth order. The individual plots show different momentum ranges, clarified by the label at the top of each graph. The color scheme corresponds to the normal form defect of the specific onion layer. The white boxes for lower normal form radii indicate a normal form defect below 10โˆ’5 . The yellow boxes denote normal form defects up to 10โˆ’4 . The orange boxes correspond to normal form defects up to 10โˆ’3 . The red boxes denote normal form defects up to 10โˆ’2.5 , and the black boxes indicate normal form defects larger than that. Each onion layer corresponds to a 0.04 ร— 0.04 box in normal form space with a thickness of 0.04% in ๐›ฟ ๐‘. . . . . 239 xxx Figure 6.50: Nonverified normal form defect for the phase space storage regions of the Muon ๐‘”-2 Storage Ring simulation with an ESQ voltage of 18.3 kV using an eleventh order map and its normal form transformation up to tenth order. The individual plots show different momentum ranges, clarified by the label at the top of each graph. The color scheme corresponds to the normal form defect of the specific onion layer. The white boxes for lower normal form radii indicate a normal form defect below 10โˆ’5 . The yellow boxes denote normal form defects up to 10โˆ’4 . The orange boxes correspond to normal form defects up to 10โˆ’3 . The red boxes denote normal form defects up to 10โˆ’2.5 , and the black boxes indicate normal form defects larger than that. Each onion layer corresponds to a 0.04 ร— 0.04 box in normal form space with a thickness of 0.04% in ๐›ฟ ๐‘. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 xxxi CHAPTER 1 INTRODUCTION Henri Poincarรฉ was a pioneer โ€“ his three volumes on โ€˜New Methods of Celestial Mechanicsโ€™ [74] were one of the greatest methodological contributions not only to the field of celestial mechanics, but also for the mathematical theory of dynamical systems in general. Numerous methods to describe and analyze dynamical systems in various research areas have been established and developed based on his work. Poincarรฉโ€™s ideas and concepts were groundbreaking, but strongly limited in their application. Performing his perturbation theory approaches by hand requires a certain simplicity or algebraic structure of the considered system. Many complex systems do not exhibit this simplicity and are impossible to solve in a purely analytic closed form. Consequently, those systems are often reduced in their complexity to ideal cases or simplified versions to solve them analytically. Computer based numerical methods have been developed to solve complex systems for very specific initial conditions with floating point accuracy. However, to develop sophisticated solutions of complex systems, which are more general than just for a specific set of initial conditions, it is critical to capture as much of the algebraic structure of the problem as possible. The differential algebra (DA) framework developed by Berz et al. [19, 15, 18, 14] (Sec. 2.1) constitutes a hybrid structure that manages both of these aspects. It captures the algebraic structure of a system up to arbitrary order to carry out the perturbation part going back to Poincarรฉโ€™s theory, while its implementation in COSY INFINITY [27, 25, 61] allows for an automated calculation of algebraic solutions in a computer environment based on floating point arithmetic. This thesis will use this powerful hybrid and its associated methods to dive into the fascinating world of nonlinear dynamical systems. The common mathematical underpinnings of many of those systems make it possible to apply the highly developed DA methods to seemingly unrelated fields of study using suitable transformations and projections. To emphasize this versatility of the methods, we analyze one problem from the field of astrodynamics in Chapter 4, and one problem from the field 1 of accelerator physics in Chapter 5. Additionally, we introduce a key technique โ€“ the DA normal form algorithm [19] โ€“ in Chapter 3, where we analyze the well known system of the centrifugal governor not in its usual linearized version, but with its high order nonlinearities. The analysis in the field of accelerator physics in Chapter 5 is concerned with the stability and the oscillation frequencies of particles in the storage ring of the Muon ๐‘”-2 Experiment at Fermilab (E989). We investigate the dependence of these frequencies on offsets in the momentum of the particles and on the amplitudes of oscillation. Nonlinear effects of the various electric field and magnetic field components of the storage rings that are used to confine the particles and bend their trajectory cause these shifts in the frequencies, which potentially influences the beamโ€™s susceptibility to resonances. In fact, for the specific ring configurations considered in this thesis, the resonance behavior and their associated fixed point structures make this analysis particularly interesting from a dynamical systems point of view. In contrast, the analysis in the field of astrodynamics in Chapter 4 is concerned with the trajectories of satellites in low and medium Earth orbits under zonal gravitational perturbation. The perturbation significantly distorts the orbits from their Keplerian form, causing them to rotate within their orbital plane and precess around the Earth at different frequencies. We present a method that elegantly solves one of the elementary challenges in astrodynamics, namely the bounded motion problem, for orbits in the Earthโ€™s zonally perturbed gravitational field. Our method generates large continuous sets of orbits, for which any two orbits remain in bounded motion for time periods of decades despite the perturbation. An essential tool in all of those applications are DA transfer maps and Poincarรฉ maps [19, 40] (see Sec. 2.2). Instead of continuously working with the equations of motions in the form of ordinary differential equations (ODE) as Poincarรฉ did, we work with maps generated from those ODEs. They yield an arbitrarily high order description of a systemโ€™s behavior between two discrete instances of time or location. Maps are particularly useful for the analysis of repetitive systems in the form of Poincarรฉ return maps, where the maps represent the systemโ€™s behavior in a chosen cross section of the motion for each turn. A repetitive application of the map to a state in that cross section 2 corresponds to the propagation of the state in the system. Accordingly, the repetitive application allows for a stroboscopic study of the repetitive motion with all the implications regarding its stability. Origin preserving Poincarรฉ return maps, which are expanded around a linearly stable fixed point, are the starting point of the DA normal form algorithm [19, 17, 16] (see Sec. 2.3). The linearly stable fixed point corresponds to a stable equilibrium state in the Poincarรฉ projection of the system. With the DA normal form algorithm, the phase space behavior around the fixed point of the map is transformed to normalized coordinates, which are closely related to action-angle coordinates. In those normal form coordinates, the phase space behavior is rotationally invariant with only amplitude dependent angle advancements up to the order of calculation. Accordingly, the angle advancements and the amplitude describe the key aspects of the dynamics straightforwardly (see Sec. 2.3.1). This generalized nonlinear normalization method up to arbitrary order is very powerful and has many applications making it the main component of many techniques used in this thesis. Chapter 3 focuses on a detailed walk through of the DA normal form algorithm using the centrifugal governor as an example. While the principal structure of the process is rather straightforward, the implications of individual steps are not always obvious. This chapter allows discussing those intricacies in full detail. One critical aspect of the normal form transformation is its sensitivity to resonances (see Sec. 2.3.2). Resonances can affect the normalization process such that the rotationally invariant structure of the resulting normal form is perturbed depending on the strength of the resonances. Hence, those resonances constitute one of the driving factors of the normal form defect (see Sec. 2.4), which is a measure of the variance of the (pseudo-)invariants produced by the normal form. This variance yields a local rate of divergence and can therefore be used as a stability estimate. Phase space regions with large normal form defects can trigger diverging phase space behavior and indicate less stable motion. As an outlook for future developments, Chapter 6 discusses the first steps of enhancing the methods for these specific applications by making them completely verified. We will see that 3 fully transferring these methods to a verified version is everything but trivial and still to be further investigated. As a starting point for the verified analysis, we introduce verified global optimization [12, 69, 29, 63, 57, 43] and its application for a verified stability estimate of the Muon ๐‘”-2 Storage Ring. The basis of this discussion and the global optimization method (see Sec. 2.6) are Taylor Models [53, 58, 54, 55, 21, 75] (see Sec. 2.5), which yield a structure for verified computations by enhancing the DA framework with rigorous remainder bounds. 4 CHAPTER 2 METHODS The methods used for this thesis are hybrids of numerical and analytical techniques based on a differential algebra (DA) framework, which was first developed to its current extent by Berz et al. [19, 14, 15]. The following summary and introduction to the DA framework (Sec. 2.1), DA maps (Sec. 2.2), and the DA normal form algorithm (Sec. 2.3) are based on [19] and have been given in similar form in my previous publications [95, 96, 93, 94]. In Sec. 2.3.1, the resulting quantities of the normal form, namely the tune, tune shifts, and normal form radii, are discussed in more detail. The influence of resonances on the normal form is described in Sec. 2.3.2. Sec. 2.4 yields an introduction to the normal form defect, a measure for the non-invariance of the normal form radii, based on [29]. The introduction to Taylor Models (Sec. 2.5) for verified computations and their applications including verified global optimization (Sec. 2.6) are based on the work of Makino and Berz et al. [53, 58, 54, 55, 29, 62]. 2.1 The Differential Algebra (DA) Framework The fundamental purpose of the DA framework [19] is to provide a mathematical backbone for computer based storage and manipulation of analytic functions. In principle, this is done by representing an analytic function ๐‘“ in terms of its Taylor polynomial expansion P ๐‘“ up to order ๐‘š, similar to how real numbers are represented by an approximation up to a certain arbitrary number of significant digits. In order to discuss the mathematical construction of the differential algebra framework in more detail, we require the notation โ€˜=๐‘š โ€™ instead of just โ€˜โ‰ˆโ€™ to clarify that both sides of such an equation are equivalent up to order ๐‘š. A Taylor polynomial expansion P ๐‘“ up to order ๐‘š represents multiple analytic functions which are equivalent up to order ๐‘š. This gives rise to the definition of equivalence classes following [19, p. 91]. The equivalence class [ ๐‘“ ] ๐‘š represents all elements ๐‘“ of the vector space of ๐‘š times 5 differentiable functions C ๐‘š (R๐‘› ) with ๐‘› real variables that have identical derivatives at the origin up to order ๐‘š. The origin is chosen out of convenience and without loss of generality โ€“ any other point may be selected. In the DA framework, the equivalence class [ ๐‘“ ] ๐‘š is represented by a DA vector, which stores all the coefficients of the Taylor expansion of ๐‘“ and the corresponding order of the terms in an orderly fashion. Operations are defined on the vector space ๐‘š ๐ท ๐‘› of all the equivalence classes [] ๐‘š . There are three operations: addition, vector multiplication, and scalar multiplication, which yield results equivalent to the result up to order ๐‘š of adding two polynomials, multiplying two polynomials, and multiplying a polynomial with a scalar. The first two operations on the equivalence classes (DA vectors) form a ring. The scalar multiplication makes the three operations on the real (or complex) DA vectors an algebra, where not every element has a multiplicative inverse. An example of such elements without a multiplicative inverse is functions without a constant part like ๐‘“ (๐‘ฅ) = ๐‘ฅ, since 1/ ๐‘“ (๐‘ฅ) = 1/๐‘ฅ is not defined at the origin and can therefore not be expanded around it. To make the algebra a differential algebra, the derivation ๐ท satisfying the Leibniz rule ๐ท ( ๐‘“ ๐‘”) = ๐‘“ ๐ท (๐‘”) + ๐‘”๐ท ( ๐‘“ )) (2.1) is introduced, which is almost trivial in the picture of differentiating polynomial expansions. The derivation opens the door to the algebraic treatment of ordinary and partial differential equations as it is common in the study of differential algebras [77, 76, 45]. Implemented in COSY INFINITY [27, 25, 61], the DA framework allows preserving the algebraic structure up to arbitrary order while manipulating the coefficients of the DA vectors with floating point accuracy. Detailed examples of the operations on 1 ๐ท 1 and 2 ๐ท 1 are given in [19] and [93], respectively. An example of a DA vector in the application of DA transfer maps and Poincarรฉ maps is given in Sec. 2.2. 2.2 DA Transfer Maps and Poincarรฉ Maps The dynamics of a system are often described by a set of ordinary differential equations (ODE) ๐‘งยฎยค = ๐‘“ (ยฎ๐‘ง, ๐‘ก), which describe the incremental change of a state ๐‘งยฎ over an independent variable ๐‘ก like 6 time. For practical purposes, it is often advantageous to generally describe the long term propagation of a state ๐‘งยฎ. In the terminology of dynamical system theory, a so-called flow operator M๐‘‡ is used to describe the action of the system on a state ๐‘งยฎ after a fixed time ๐‘‡. Since it is often impossible to determine the flow in a closed form, numerical integration of the ODE is required. The DA framework allows for a hybrid integration that conserves the algebraic structure up to arbitrary order during the integration. Integrating a local expansion ๐›ฟยฎ๐‘ง ini around an initial state ๐‘งยฎ0 yields the final state ๐‘งยฎfin in form of an ๐‘š order flow map M๐‘‡ , which depends on the expansion in (๐›ฟยฎ๐‘งini , ๐›ฟ๐œ‚), ยฎ where ๐›ฟ๐œ‚ยฎ is the expansion around a reference set of parameters ๐œ‚ยฎ0 . More generally speaking, a transfer map M algebraically expresses how a final state ๐‘งยฎfin is dependent on an initial state ๐‘งยฎini and system parameters ๐œ‚, ยฎ as ๐‘งยฎfin = M (ยฎ๐‘งini , ๐œ‚) ยฎ . (2.2) Transfer maps are also called propagators or simply maps. The expansion point of the map belongs to a chosen reference orbit/state of the system, e.g. a (pseudo-)closed orbit for a fixed point map and/or the ideal orbit of the unperturbed system. There are special transfer maps called Poincarรฉ maps [74] that constrain the initial and final state to Poincarรฉ surfaces Sini and Sfin , respectively. For the simulation of storage rings and their particle optical elements, this concept is used to represent how the state after a storage ring element depends on system parameters and the state before the element. A setup of multiple consecutive storage ring elements is described by the composition of their Poincarรฉ maps. Poincarรฉ return maps represent the case where Sini is equal to Sfin . They are particularly useful for the representation of dynamics in repetitive systems like the ones considered in this thesis. Multiple applications of a Poincarรฉ return map correspond to the propagation of the system. The Poincarรฉ return maps are particularly advantageous when they are origin preserving, i.e., the expansion point is a fixed point of the map, because system dynamics represented by origin preserving Poincarรฉ return maps can be further analyzed by normal form methods and for the asymptotic stability of the system. 7 Constraining the map to the Poincarรฉ surface S is often done by calculating the flow of an ODE and projecting it onto the surface S. This reduces the dimension of the original map and generates the Poincarรฉ map. An implementation of a timewise projection onto a surface S defined ยฎ = 0 is outlined in [40]. by ๐œŽ(ยฎ๐‘ง, ๐œ‚) The projection uses DA inversion methods that compute the inverse Aโˆ’1 to the auxiliary map A, which contains the constraining conditions of the Poincarรฉ surface S. Given that A has no constant part, the auxiliary map and its inverse satisfy Aโˆ’1 โ—ฆ A =๐‘š A โ—ฆ Aโˆ’1 =๐‘š I . (2.3) The basic idea of the projection of a transfer map M onto a surface defined by ๐œŽ(ยฎ๐‘ง, ๐œ‚)ยฎ = 0 is to replace one of the variables or parameters of M by an expression in terms of all the other variables and parameters such that the constraint ๐œŽ( M) = 0 is satisfied. This eliminates the corresponding component of the map and thereby reduces its dimensionality. In [40], the timewise projection is prepared by calculating an expansion of the map M in time ๐‘ก. The DA inversion methods are then used to find the intersection time ๐‘กโ˜… (ยฎ๐‘ง, ๐œ‚) ยฎ dependent on the state variables ๐‘งยฎ and system parameters ๐œ‚ยฎ such that ๐œŽ( M (ยฎ๐‘ง, ๐œ‚, ยฎ ๐‘กโ˜… (ยฎ๐‘ง, ๐œ‚))) ยฎ = 0. (2.4) 2.3 The DA Normal Form Algorithm The DA normal form algorithm [19] is an advancement from the DA-Lie based version, the first arbitrary order algorithm by Forest, Berz, and Irwin [38]. Given an origin preserving map M of a repetitive Hamiltonian system, where the components of the map are in phase space coordinates, the DA normal form algorithm provides a nonlinear change of the phase space variables by an order-by-order transformation to rotationally invariant normal form coordinates. Implemented in COSY INFINITY [27, 25, 61], this is a fully automated process, which can be performed up to arbitrary order. It is only limited by floating point accuracy and the capability of the computer system to handle DA vectors of the chosen computation order ๐‘š calc and dimension. 8 In the standard configuration, order ten calculations of a three dimensional system (with six phase space variables) are easily manageable. In Chapter 3, the normal form algorithm is explained in great detail for the one dimensional system (with two phase space variables) of a centrifugal governor. Here we want to explain the more general form for a 2๐‘› dimensional symplectic system with an optional parameter dependence on ๐‘›๐œ‚ parameters summarized in ๐œ‚. ยฎ The explanations are largely based on [19]. For parameter dependent maps, the algorithm starts by expanding the origin preserving map M (ยฎ๐‘ง, ๐œ‚) ยฎ around its parameter dependent fixed point ๐‘งยฎPDFP ( ๐œ‚), ยฎ which satisfies M (ยฎ๐‘งPDFP ( ๐œ‚), ยฎ ๐œ‚)ยฎ = ๐‘งยฎPDFP ( ๐œ‚). ยฎ (2.5) Defining the extended map N = ( M โˆ’ I๐‘งยฎ, ๐œ‚), ยฎ the parameter dependent fixed point ๐‘งยฎPDFP is determined by evaluating the inverse of N at the expansion point ๐‘งยฎ = 0: ยฎ   (ยฎ๐‘งPDFP ( ๐œ‚) ยฎ = N โˆ’1 0, ยฎ , ๐œ‚) ยฎ ๐œ‚ยฎ . (2.6) The map M is then expanded around its parameter dependent fixed point ๐‘งยฎPDFP . The resulting map M0 = L + ๐‘š U๐‘š consists of a linear part L and the nonlinear parts U๐‘š of ร order ๐‘š. Due to the transformation to the parameter dependent fixed point, the map has no terms that only depend on parameters. Accordingly, the entire linear part is independent of parameters. The variables of the map are the canonical phase space coordinates ๐‘งยฎ = ( ๐‘žยฎ0 , ๐‘ยฎ0 ) and, if applicable, parameters ๐œ‚.ยฎ The normal form algorithm transforms this map order by order up to the full calculation order ๐‘š calc of the map. For each order ๐‘š, the transformation step has the following form M๐‘š = A๐‘š โ—ฆ M๐‘šโˆ’1 โ—ฆ Aโˆ’1 ๐‘š , (2.7) where A๐‘š is the transformation map (also just called transformation) and Aโˆ’1 ๐‘š is its inverse. The result of the ๐‘šth order transformation step is the map M๐‘š . Hence, M๐‘šโˆ’1 is the result from the previous transformation step or M0 from above. The last transformation step is for order ๐‘š = ๐‘š calc . The first step of the algorithm, for ๐‘š = 1, is to linearly decouple the map into ๐‘› two dimen- sional subspaces. The linear transformation diagonalizes the system, transforming the (parameter 9 dependent) fixed point map into the complex conjugate eigenvector space of its linear part. We assume linearly stable behavior around the (parameter dependent) fixed point of the map with distinct complex conjugate eigenvalue pairs of magnitude one since this property is shared among all systems considered in this thesis (see [19] for other cases). If any of the eigenvalues ๐œ†โ˜… had an absolute value larger than 1, the motion would be unstable since the state on the corresponding eigenvector ๐‘ฃยฎโ˜… would grow in magnitude by a factor of |๐œ†โ˜… | > 1 with each iteration. Additionally, eigenvalues of symplectic maps come in reciprocal pairs such that eigenvalues with a magnitude smaller than 1 have a reciprocal partner eigenvalue |๐œ†โ˜… | > 1, which are again linearly unstable. ยฑ๐‘–๐œ‡ ๐‘— The complex conjugate eigenvalue pairs ๐‘’ of the diagonalized linear part are grouped together such that the matrix ๐‘…ห† of the diagonalized linear part R has the following decoupled form ยฉ ๐‘…ห†1 ยช ... ยฉ๐‘’ +๐‘–๐œ‡ ๐‘— ยญ ยฎ ยญ ยฎ 0 ยช ห† ๐‘…=ยญ ยญ ๐‘…ห†๐‘™ ยฎ ยฎ where ห† ๐‘… ๐‘— = ยญยญ ยฎ. (2.8) .. โˆ’๐‘–๐œ‡ ๐‘— ยฎ 0 ยญ ยฎ ยญ . ยฎ ๐‘’ ยญ ยฎ ยซ ยฌ ยซ ๐‘…ห† ๐‘› ยฌ The resulting map of the first transformation step โ€“ linear transformation โ€“ is M1 = R + ๐‘š S๐‘š , ร where the new nonlinear terms of order ๐‘š that resulted from the linear transformation are denoted by S๐‘š . The complex phase ยฑ๐œ‡ ๐‘— of the eigenvalue pairs will be of critical importance in the nonlinear transformations of the algorithm. In summary, the first transformation step performed the following operation โˆ‘๏ธ โˆ‘๏ธ M1 = A1 โ—ฆ M0 โ—ฆ Aโˆ’1 1 = A1 โ—ฆ L โ—ฆ A1 + โˆ’1 A1 โ—ฆ U๐‘š โ—ฆ Aโˆ’1 1 =R+ ๐‘†๐‘š , (2.9) ๐‘š ๐‘š where A1 is the linear transformation from the original coordinate space ( ๐‘žยฎ0 , ๐‘ยฎ0 ) to the complex conjugate coordinate space ( ๐‘žยฎ1 , ๐‘ยฎ1 ) and Aโˆ’1 1 is its inverse for the transformation in the opposite direction. With the linearly decoupled map, the following steps of the normal form algorithm can be performed for each of these linearly decoupled subspaces separately. The ๐‘—th subspace of the linearly 10 decoupled map M1 can be explicitly written as โˆ‘๏ธ ยฉ๐‘’ +๐‘–๐œ‡ ๐‘— 0 ยช ยฉ ๐‘ž 1, ๐‘— ยช M1, ๐‘— ( ๐‘žยฎ1 , ๐‘ยฎ1 , ๐œ‚) ยฎ = R๐‘— + S๐‘š, ๐‘— = ยญยญ ยฎยญ ยฎ โˆ’๐‘–๐œ‡ ๐‘— ยฎ ยญ ยฎ ๐‘š 0 ๐‘’ ๐‘ 1, ๐‘— ยซ ยฌยซ ยฌ ยฉS + ยช ๐‘› ๐‘›๐œ‚ ยญ ๐‘š( ๐‘˜ยฎ+ , ๐‘˜ยฎโˆ’ , ๐‘˜ยฎ๐œ‚ ), ๐‘— ยฎ ร– ๐‘˜+ ๐‘˜โˆ’ ๐œ‚ โˆ‘๏ธ ร– + ยญ ยฎ (๐‘ž 1,๐‘™ ) ๐‘™ ( ๐‘ 1,๐‘™ ) ๐‘™ (๐œ‚๐‘ข ) ๐‘˜ ๐‘ข , (2.10) ยญ โˆ’ ๐‘š=|| ๐‘˜ยฎ + + ๐‘˜ยฎ โˆ’ || 1 +|| ๐‘˜ยฎ ๐œ‚ || 1 S๐‘š( ๐‘˜ยฎ + , ๐‘˜ยฎ โˆ’ , ๐‘˜ยฎ ๐œ‚ ), ๐‘— ๐‘™=1 ยฎ ๐‘ข=1 ยซ ยฌ where ๐‘˜ ๐‘™+ represents the positive integer exponent of ๐‘ž 1,๐‘™ , ๐‘˜ ๐‘™โˆ’ represents the positive integer exponent ๐œ‚ of ๐‘ 1,๐‘™ , and ๐‘˜ ๐‘ข represents the positive integer exponent of ๐œ‚๐‘ข . The positive integer exponents are summarized in the vectors ๐‘˜ยฎ + , ๐‘˜ยฎ โˆ’ , and ๐‘˜ยฎ ๐œ‚ , respectively. The ๐ฟ 1 -Norm || ยท || 1 of the sum of these vectors is used to ensure that only polynomial terms of order ๐‘š are considered. To better understand the expression in Eq. (2.10), we present some terms of the Mโˆ’ 1, ๐‘— component โˆ’๐‘–๐œ‡ Mโˆ’ ยฎ = ๐‘’ ๐‘— ยท ๐‘ 1, ๐‘— + S โˆ’ 1, ๐‘— ( ๐‘žยฎ1 , ๐‘ยฎ1 , ๐œ‚)  ยท ๐‘ž 21,1 + ... (2.11) 2 (2,0,...,0)๐‘‡ ,(0,...,0)๐‘‡ ,(0,...,0)๐‘‡ , ๐‘— + S โˆ’  ยท ๐‘ž 1, ๐‘— ๐‘ 1,๐‘™ + ... 2 (0,...,0,๐‘˜ + =1,0,...,0)๐‘‡ ,(0,...,0,๐‘˜ โˆ’ =1,0,...,0)๐‘‡ ,(0,...,0)๐‘‡ , ๐‘— ๐‘— ๐‘™ + S โˆ’  ยท ๐‘ 1,๐‘› ๐œ‚1 + ... 2 (0,...,0)๐‘‡ ,(0,...,0,1)๐‘‡ ,(1,0,...,0)๐‘‡ , ๐‘— Due to the linear transformation into the complex conjugate eigenvector space of the purely real linear part, the two components of each subspace form a complex conjugate pair. The โ€˜+โ€™ and โ€˜-โ€™ notation is used, where the sign corresponds to the sign of the complex eigenvalue phase of the map component of that subspace. Specifically, this means that โˆ’ โˆ’ M+1, ๐‘— = M1, ๐‘— with ๐‘ž 1, ๐‘— = ๐‘ 1, ๐‘— and S + ยฎ+ ยฎโˆ’ ยฎ๐œ‚ = S ๐‘š( ๐‘˜ยฎ+ , ๐‘˜ยฎโˆ’ , ๐‘˜ยฎ๐œ‚ ), ๐‘— . (2.12) ๐‘š( ๐‘˜ , ๐‘˜ , ๐‘˜ ), ๐‘— This property is maintained throughout all the following nonlinear transformation steps, which are performed order by order starting with order two. The general form of the nonlinear transformation is A๐‘š =๐‘š I + T๐‘š , where T๐‘š is a polynomial containing only terms of order ๐‘š. Hence, the transformation A๐‘š is a near-identity transformation and a full identity up to order ๐‘š โˆ’ 1. The transformation A๐‘š is determined by finding T๐‘š such that the ๐‘šth order of the resulting map M๐‘š 11 is simplified or even eliminated when the transformation A๐‘š and its inverse Aโˆ’1 ๐‘š =๐‘š I โˆ’ T๐‘š are applied to M๐‘šโˆ’1 in the ๐‘šth order nonlinear transformation step (see Eq. (2.7)). The higher order terms of the transformation A๐‘š do not influence the ๐‘šth order terms of the map. Hence, they are irrelevant for the ๐‘šth order transformation step and can be chosen freely, e.g. to make the transformation symplectic with A๐‘š = exp(๐ฟ T๐‘š ) which we will do (see [19]). However, the higher orders of the resulting map M๐‘š are strongly dependent on A๐‘š , its higher order terms, and its corresponding inverse. In Chapter 3, the influences of the second order transformation on the third order terms of the resulting map are analyzed in great detail. While these influences are not to be dismissed, the key element of this ๐‘šth order transformation step is the elimination of as many ๐‘šth order terms of the resulting map M๐‘š as possible by a smart choice of T๐‘š . Given the map M๐‘šโˆ’1 , representing M simplified up to order ๐‘š โˆ’ 1 and applying A๐‘š and its inverse to it, yields [19, Eq. (7.60)]: A๐‘š โ—ฆ M๐‘šโˆ’1 โ—ฆ Aโˆ’1 ๐‘š =๐‘š ( I + T๐‘š ) โ—ฆ ( R + S๐‘š ) โ—ฆ ( I โˆ’ T๐‘š ) =๐‘š ( I + T๐‘š ) โ—ฆ ( R โˆ’ R โ—ฆ T๐‘š + S๐‘š ) =๐‘š R + S๐‘š + [ T๐‘š , R] , (2.13) where R is the diagonalized linear part and S๐‘š represents only the ๐‘šth order terms of the map M๐‘šโˆ’1 (the leading order of terms that have not been simplified yet). The equations above only consider terms up to order ๐‘š, since terms of order ๐‘š + 1 and larger are irrelevant for determining T๐‘š . The maximum simplification would be achieved by finding T๐‘š such that the commutator C๐‘š = T๐‘š โ—ฆ R โˆ’ R โ—ฆ T๐‘š = [T๐‘š , R] = โˆ’S๐‘š , (2.14) which would eliminate all nonlinear terms S๐‘š of order ๐‘š. Since the commutator only involves T๐‘š and R we can investigate this transformation separately in the ๐‘› individual subspaces. The components of the ๐‘—th subspace of the commutator C๐‘š = [ T๐‘š , R] 12 are ยฉC + ยช ๐‘› ๐‘›๐œ‚ ยญ ๐‘š( ๐‘˜ยฎ+ , ๐‘˜ยฎโˆ’ , ๐‘˜ยฎ๐œ‚ ), ๐‘— ยฎ ร– ๐‘˜+ ๐‘˜โˆ’ ๐œ‚ โˆ‘๏ธ ร– C๐‘š, ๐‘— = ยญ ยฎ (๐‘ž ๐‘™ ) ๐‘™ ( ๐‘ ๐‘™ ) ๐‘™ (๐œ‚๐‘ข ) ๐‘˜ ๐‘ข , (2.15) ยญ โˆ’ ๐‘š=|| ๐‘˜ยฎ + + ๐‘˜ยฎ โˆ’ || 1 +|| ๐‘˜ยฎ ๐œ‚ || 1 C๐‘š( ๐‘˜ยฎ + , ๐‘˜ยฎ โˆ’ , ๐‘˜ยฎ ๐œ‚ ), ๐‘— ๐‘™=1 ยฎ ๐‘ข=1 ยซ ยฌ where   ! ๐‘– ๐œ‡ยฎ ๐‘˜ยฎ + โˆ’ ๐‘˜ยฎ โˆ’ ยฑ๐‘–๐œ‡ ๐‘— C ยฑ ยฎ+ ยฎโˆ’ ยฎ๐œ‚ = T ยฑ ยฎ+ ยฎโˆ’ ยฎ๐œ‚ ๐‘’ โˆ’๐‘’ . (2.16) ๐‘š( ๐‘˜ , ๐‘˜ , ๐‘˜ ), ๐‘— ๐‘š( ๐‘˜ , ๐‘˜ , ๐‘˜ ), ๐‘— Accordingly, the commutator terms C ยฑ ยฎ+ ยฎโˆ’ ยฎ๐œ‚ can eliminate their corresponding nonlinear ๐‘š( ๐‘˜ , ๐‘˜ , ๐‘˜ ), ๐‘— ยฑ terms of the map S ยฎ+ ยฎโˆ’ ยฎ๐œ‚ by choosing ๐‘š( ๐‘˜ , ๐‘˜ , ๐‘˜ ), ๐‘— โˆ’S ยฑ ยฎ+ ยฎโˆ’ ยฎ๐œ‚ ๐‘š( ๐‘˜ , ๐‘˜ , ๐‘˜ ), ๐‘— T ยฑ ยฎ+ ยฎโˆ’ ยฎ๐œ‚ =   , (2.17) ๐‘š( ๐‘˜ , ๐‘˜ , ๐‘˜ ), ๐‘— ๐‘– ๐œ‡ยฎ ๐‘˜ยฎ + โˆ’ ๐‘˜ยฎ โˆ’ ยฑ๐‘–๐œ‡ ๐‘— ๐‘’ โˆ’๐‘’ if   ๐‘– ๐œ‡ยฎ ๐‘˜ยฎ + โˆ’ ๐‘˜ยฎ โˆ’ ยฑ๐‘–๐œ‡ ๐‘— ๐‘’ โˆ’๐‘’ โ‰  0. (2.18) In other words, only the S ยฑ ยฎ+ ยฎโˆ’ ยฎ๐œ‚ terms corresponding to C ยฑ ยฎ+ ยฎโˆ’ ยฎ๐œ‚ for which the ๐‘š( ๐‘˜ , ๐‘˜ , ๐‘˜ ), ๐‘— ๐‘š( ๐‘˜ , ๐‘˜ , ๐‘˜ ), ๐‘— condition (see [19, Eq. (7.65)]) โˆ‘๏ธ   mod2๐œ‹ ยญ ๐œ‡ ๐‘— (๐‘˜ +๐‘— โˆ’ ๐‘˜ โˆ’๐‘— โˆ“ 1) + ๐œ‡๐‘™ ๐‘˜ ๐‘™+ โˆ’ ๐‘˜ ๐‘™โˆ’ ยฎ = 0, (2.19) ยฉ ยช ยซ ๐‘™โ‰  ๐‘— ยฌ is satisfied, survive. A straightforward solution of the condition in Eq. (2.19) is ๐‘˜ +๐‘— โˆ’ ๐‘˜ โˆ’๐‘— = ยฑ1 โˆง ๐‘˜ ๐‘™+ = ๐‘˜ ๐‘™โˆ’ โˆ€๐‘™ โ‰  ๐‘—, (2.20) where the first condition concerns the ๐‘—th subspace and the second condition is regarding all the other subspaces ๐‘™ with ๐‘™ โ‰  ๐‘—. The surviving terms of the ๐‘šth order transformation step in the ๐‘—th subspace can be generally written as S+ ยฎ ยฎ ๐‘˜ยฎ ๐œ‚ ), ๐‘— and S โˆ’ ยฎ ยฎ with 2|| ๐‘˜ยฎ || 1 + 1 + || ๐‘˜ยฎ ๐œ‚ || 1 = ๐‘š, (2.21) ๐‘š( ๐‘˜+๐‘’ยฎ ๐‘— , ๐‘˜, ๐‘š( ๐‘˜, ๐‘˜+๐‘’ยฎ ๐‘— , ๐‘˜ยฎ ๐œ‚ ), ๐‘— where the unit vector ๐‘’ยฎ ๐‘— consists only of zeros except for a 1 at the ๐‘—th entry. 13 From Eq. (2.21) it becomes clear that only certain terms of uneven order in the phase space coordinates ( ๐‘ž,ยฎ ๐‘) ยฎ survive. These terms have the special property that each complex conjugate phase space variable pair is raised to the same exponent except for the phase space variable pair of the respective subspace. So, all even order terms in phase space coordinates can be eliminated by the nonlinear normal form transformations. The remaining terms of S๐‘š (from Eq. (2.21)) describe the entire dynamics of the systems in a nutshell and are the key elements of the normal form and therefore essential for further analysis of the dynamics. Resonances between the complex phases ๐œ‡ยฎ of the different subspaces in the denominator of Eq. (2.17) can break this special structure and therefore the rotational invariance of the normal form as will be discussed in Sec. 2.3.2. For now, we will continue only with the terms that are supposed to survive, namely the terms specified in Eq. (2.21). Once the nonlinear transformation steps transformed the map up to its full order ๐‘š = ๐‘š calc , the map has been significantly simplified to ยฉM+๐‘š, ๐‘— ยช ยฉ ๐‘ž ๐‘š, ๐‘— ๐‘“ ๐‘—+ ๐‘ž ๐‘š,1 ๐‘ ๐‘š,1 , ๐‘ž ๐‘š,2 ๐‘ ๐‘š,2 , ..., ๐‘ž ๐‘š,๐‘› ๐‘ ๐‘š,๐‘› , ๐œ‚ยฎ ยช  ยญ โˆ’ ยฎ=ยญ ยญ ยฎ ยญ ยฎ , ยฎ (2.22) M ๐‘ โˆ’ ๐‘“ ๐‘— ๐‘ž ๐‘š,1 ๐‘ ๐‘š,1 , ๐‘ž ๐‘š,2 ๐‘ ๐‘š,2 , ..., ๐‘ž ๐‘š,๐‘› ๐‘ ๐‘š,๐‘› , ๐œ‚ยฎ ยซ ๐‘š, ๐‘— ยฌ ยซ ๐‘š, ๐‘— ยฌ where ๐‘› ๐‘›๐œ‚ โˆ‘๏ธ ร– ๐‘˜ ร– ๐œ‚ ๐‘“ ๐‘—+ = ๐‘’ +๐‘–๐œ‡ + S+ ยฎ ยฎ ๐‘˜ยฎ ๐œ‚ ), ๐‘— ๐‘ž ๐‘š,๐‘™ ๐‘ ๐‘š,๐‘™ ๐‘™ (๐œ‚๐‘ข ) ๐‘˜ ๐‘ข . (2.23) ๐‘š( ๐‘˜+๐‘’ยฎ ๐‘— , ๐‘˜, ๐‘š=2|| ๐‘˜ยฎ || 1 +1+|| ๐‘˜ยฎ ๐œ‚ || 1 ๐‘™=1 ๐‘ข=1 Since the original map is real, the last step of the algorithm is transforming the resulting map to the real normal form basis ( ๐‘žยฎNF , ๐‘ยฎNF ), which is composed of the real and imaginary parts of the current complex conjugate basis ( ๐‘žยฎ๐‘š , ๐‘ยฎ๐‘š ). The relation between the bases is ๐‘ž ๐‘š, ๐‘— + ๐‘ ๐‘š, ๐‘— ๐‘ž ๐‘š, ๐‘— โˆ’ ๐‘ ๐‘š, ๐‘— ๐‘ž NF, ๐‘— = , ๐‘ NF, ๐‘— = , and (2.24) 2 2๐‘– ๐‘ž ๐‘š, ๐‘— = ๐‘ž NF, ๐‘— + ๐‘– ๐‘ NF, ๐‘— , ๐‘ ๐‘š, ๐‘— = ๐‘ž NF, ๐‘— โˆ’ ๐‘– ๐‘ NF, ๐‘— . (2.25) The squared normal form radius ๐‘Ÿ NF, 2 ๐‘— is given by the product of ๐‘ž ๐‘š, ๐‘— ๐‘ ๐‘š, ๐‘— , with ๐‘ž ๐‘š, ๐‘— ๐‘ ๐‘š, ๐‘— = ๐‘ž 2NF, ๐‘— + ๐‘ 2NF, ๐‘— = ๐‘Ÿ NF, 2 ๐‘—. (2.26) 14 Applying the basis transformation to the map components of M๐‘š in each subspace yields MNF, ๐‘— = Areal, ๐‘— โ—ฆ M๐‘š, ๐‘— โ—ฆ Aโˆ’1 real, ๐‘—    + 2 2 1 ยฉ 1 1ยชยฎ ยฉยญ ๐‘“ ๐‘— ๐‘Ÿ NF,1 , ..., ๐‘Ÿ NF,๐‘› , ๐œ‚ยฎ ๐‘ž NF, ๐‘— + ๐‘– ๐‘ NF, ๐‘— ยชยฎ = ยญยญ ยท   ยฎ 2 โˆ’๐‘– ๐‘– ยฎ ยญ ๐‘“ โˆ’ ๐‘Ÿ 2 , ..., ๐‘Ÿ 2 , ๐œ‚ยฎ ๐‘ž NF,๐‘› NF, ๐‘— โˆ’ ๐‘– ๐‘ NF, ๐‘— ยซ ยฌ ยซ ๐‘— NF,1 ยฌ     ยฉ 21 ๐‘“ ๐‘—+ + ๐‘“ยฏ๐‘—+ ๐‘ž NF, ๐‘— + 2๐‘– ๐‘“ ๐‘—+ โˆ’ ๐‘“ยฏ๐‘—+ ๐‘ NF, ๐‘— ยช = ยญยญ     ยฎ โˆ’๐‘– ๐‘“ + โˆ’ ๐‘“ยฏ+ ๐‘ž 1 + ยฏ + ยฎ 2 ๐‘— ๐‘— NF, ๐‘— + 2 ๐‘“ ๐‘— + ๐‘“ ๐‘— ๐‘ NF, ๐‘— ยซ ยฌ       ยฉRe ๐‘“ ๐‘—+ ๐‘Ÿ NF,1 2 2 , ..., ๐‘Ÿ NF,๐‘› , ๐œ‚ยฎ โˆ’Im ๐‘“ ๐‘—+ ๐‘Ÿ NF,1 2 2 , ..., ๐‘Ÿ NF,๐‘› , ๐œ‚ยฎ ยช ยฉ ๐‘ž NF, ๐‘— ยช = ยญยญ       ยฎยฎ ยท ยญยญ ยฎ. (2.27) + 2 2 + 2 2 ยฎ Im ๐‘“ ๐‘— ๐‘Ÿ NF,1 , ..., ๐‘Ÿ NF,๐‘› , ๐œ‚ยฎ Re ๐‘“ ๐‘— ๐‘Ÿ NF,1 , ..., ๐‘Ÿ NF,๐‘› , ๐œ‚ยฎ ๐‘ NF, ๐‘— ยซ ยฌ ยซ ยฌ Writing ๐‘“ ๐‘—+ and its complex conjugate counterpart ๐‘“ ๐‘—โˆ’ in terms of complex phases with     ยฑ๐‘–ฮ› ๐‘Ÿ 2 ,...,๐‘Ÿ 2 , ยฎ ๐œ‚ ๐‘“ ๐‘—ยฑ ๐‘Ÿ NF,12 2 , ..., ๐‘Ÿ NF,๐‘› , ๐œ‚ยฎ = ๐‘’ ๐‘— NF,1 NF,๐‘› (2.28) yields the following normal form       2 ยฉcos ฮ› ๐‘— ๐‘Ÿ NF,1 2 , ..., ๐‘Ÿ NF,๐‘› , ๐œ‚ยฎ โˆ’ sin ฮ› ๐‘— ๐‘Ÿ NF,1 2 2 , ..., ๐‘Ÿ NF,๐‘› , ๐œ‚ยฎ ยช ยฉ ๐‘ž NF, ๐‘— ยช MNF, ๐‘— = ยญยญ       ยฎยฎ ยท ยญยญ ยฎ , (2.29) 2 2 2 2 ยฎ sin ฮ› ๐‘— ๐‘Ÿ NF,1 , ..., ๐‘Ÿ NF,๐‘› , ๐œ‚ยฎ cos ฮ› ๐‘— ๐‘Ÿ NF,1 , ..., ๐‘Ÿ NF,๐‘› , ๐œ‚ยฎ ๐‘ NF, ๐‘— ยซ ยฌ ยซ ยฌ which clearly shows the circular phase space behavior in normal form subspaces with only amplitude   ๐‘ŸยฎNF,sqr = ๐‘Ÿ NF,1 2 2 , ..., ๐‘Ÿ NF,๐‘› (2.30) and parameter ๐œ‚ยฎ depended angle advancements ฮ›. ยฎ The radii of the circular motion โ€“ the normal form radii โ€“ are constants of motion up to the calculation order. The entire dynamics in the normal form are given by the constant angle advancements ฮ› ยฎ along the circular phase space curves. The rotational invariance implies an interpretation of the normal form as an averaged representation of the original Poincarรฉ return map M, in the limit where the map application is repeated infinitely many times. Normalizing the angle advancements ฮ› ยฎ to [0, 1] yields the tunes ๐œˆยฎ and amplitude and parameter dependent tune shifts ๐›ฟ ๐œˆยฎ (ยฎ ๐‘Ÿ NF,sqr , ๐œ‚). ยฎ Accordingly, ยฎ ๐‘Ÿ NF,sqr , ๐œ‚) ฮ›(ยฎ ยฎ = ๐œˆยฎ + ๐›ฟ ๐œˆยฎ (ยฎ ยฎ ๐‘Ÿ NF,sqr , ๐œ‚). (2.31) 2๐œ‹ 15 The normal form transformation A and its inverse Aโˆ’1 are given by the composition of all the individual transformations of each transformation step with MNF = Areal โ—ฆ A๐‘š โ—ฆ A๐‘šโˆ’1 โ—ฆ ... โ—ฆ A1 โ—ฆM โ—ฆ Aโˆ’1 โˆ’1 โˆ’1 โˆ’1 1 โ—ฆ ... โ—ฆ A๐‘šโˆ’1 โ—ฆ A๐‘š โ—ฆ Areal . (2.32) | {z } | {z } A Aโˆ’1 The normal form transformation A yields how the normal form variables (๐‘ž NF, ๐‘— , ๐‘ NF, ๐‘— ) depend on the original phase space variables ( ๐‘žยฎ0 , ๐‘ยฎ0 ) and, if considered, system parameters ๐œ‚, ยฎ which suggests the following notation for A and its inverse A = ( ๐‘žยฎNF ( ๐‘žยฎ0 , ๐‘ยฎ0 , ๐œ‚) ยฎ , ๐‘ยฎNF ( ๐‘žยฎ0 , ๐‘ยฎ0 , ๐œ‚)) ยฎ (2.33) Aโˆ’1 = ( ๐‘žยฎ0 ( ๐‘žยฎNF , ๐‘ยฎNF , ๐œ‚)ยฎ , ๐‘žยฎ0 ( ๐‘žยฎNF , ๐‘ยฎNF , ๐œ‚)) ยฎ . (2.34) 2.3.1 Tunes, Tune Shifts, and Normal Form Radii DA normal form methods are used to transform the origin preserving phase space Poincarรฉ return map to the rotationally invariant normal form up to calculation order. From the normal form, the angle advancements ฮ›(ยฎ ยฎ ๐‘Ÿ NF,sqr , ๐œ‚) ยฎ as a functions of amplitude ๐‘ŸยฎNF,sqr and parameters ๐œ‚ยฎ are particularly straightforward to extract. Scaling the angle advancements in each of the normal form phase spaces to [0, 1] instead of [0, 2๐œ‹] provides the average number of phase space revolutions per system revolution represented by the Poincarรฉ return map. In beam physics terminology, the frequencies of normal form phase space revolutions is known as the tunes ๐œˆยฎ and their amplitude and parameter dependent tune shifts ๐›ฟ ๐œˆยฎ (ยฎ ยฎ [19]. ๐‘Ÿ NF,sqr , ๐œ‚) The tune ๐œˆ ๐‘— corresponds to the scaled complex phase ๐œ‡ ๐‘— of the complex conjugate eigenvalues ๐œ†ยฑ๐‘— of the linear transformation. Hence, the tune is related to the linear motion around the expansion point, i.e., the motion โ€˜infinitely closeโ€™ to the expansion point. Interpreting the tune and its tune shifts as the phase space rotation frequency suggests that the tune โ€“ the phase space rotation frequency of the expansion point โ€“ is a rotation with no amplitude, where the frequency is determined by the linear motion around the expansion point. In particular, this means that different maps with the 16 same expansion point can have different tunes depending on the linear motion around the expansion point. Since the tunes are calculated from the linear coefficients directly without any nonlinear transformations, performing the tune calculation with parameter dependent linear coefficients directly yields the parameter dependent tune shifts. The tune shifts indicate the change of the phase space rotation frequency dependent on the phase space amplitudes ๐‘Ÿ NF, ๐‘— and variations in the system parameters ๐œ‚. ยฎ Since the normal form transformation is symplectic, it preserves the phase space volume, which is critical to understanding the connection between the original phase space coordinates and their normal form radii. If the system is only weakly coupled between the different phase spaces, the normal form radius ๐‘Ÿ NF, ๐‘— is a measure for the invariant phase space area of the ๐‘—th subspace denoted by ๐ด ๐‘— . Hence, the original phase space coordinates of an invariant phase space orbit in the ๐‘—th subspace enclose the area ๐ด ๐‘— , โˆš๏ธ which roughly corresponds to the normal form radius of ๐‘Ÿ NF, ๐‘— = ๐ด ๐‘— /๐œ‹. The normal form radii are the link between the tune dependencies and the original coordinates. The dependency of the tune shifts on the normal form radii is a result of the surviving terms S๐‘š of the nonlinear normal form transformations. However, the crucial terms are the T๐‘š terms from Eq. (2.17) that are used to cancel all the other nonlinear terms S๐‘š . On the one hand, the T๐‘š terms determine how the original coordinates ๐‘งยฎ = ( ๐‘ž, ยฎ ๐‘) ยฎ and the system parameters ๐œ‚ยฎ relate to the normal form radii ๐‘Ÿ NF, ๐‘— , since the T๐‘š are the essential part of the normal form transformation. On the other hand, they influence the higher order nonlinear terms S๐‘™ with ๐‘™ > ๐‘š, which either survive and determine the dependency of the tune shifts on the normal form radii, or they determine the higher order terms T๐‘™ . 2.3.2 Resonances The denominator of T๐‘š in Eq. (2.17) has a potentially large effect on the size of T๐‘š the closer it is to satisfying the resonance condition in Eq. (2.19). If the condition is satisfied, the corresponding nonlinear terms in S๐‘š cannot be eliminated. Accordingly, terms survive which do not fit the normal form structure. They break the normal form by the size of their respective coefficient. 17 If the condition is almost satisfied close to a resonance, then the denominator of T๐‘š becomes very small, making T๐‘š very large. In this situation, there are two options. One option is to continue the procedure with the very large T๐‘š coefficient, which conserves the normal form structure but yields diverging coefficients in all higher order terms. The other option is to let the corresponding term in S๐‘š survive, which breaks the normal form structure but avoids a divergence of the coefficients. In practice, one chooses a cutoff value for the size of the denominator, which restricts the size of potentially diverging coefficients. If the denominator is smaller than the cutoff value, the T๐‘š coefficient is set to zero, letting the corresponding S๐‘š term survive. Rewriting the resonance condition in terms of tunes yields ๐‘คยฎ ยท ๐œˆยฎ = ๐‘”, (2.35) where ๐‘คยฎ consists only of integer values and ๐‘” is a natural number N0 . The values in ๐‘คยฎ and ๐‘” are chosen such that the greatest common divisor of all values is 1. With this definition, the order of the resonance is given by ๐‘š res = || ๐‘ค||ยฎ 1. In the normal form algorithm a tune resonance defined by ( ๐‘ค, ยฎ ๐‘”) appears in all terms S ยฑ ยฎ+ ยฎโˆ’ ยฎ๐œ‚ for which ๐‘š( ๐‘˜ , ๐‘˜ , ๐‘˜ ), ๐‘— ๐‘ค ๐‘— = ๐‘˜ +๐‘— โˆ’ ๐‘˜ โˆ’๐‘— โˆ“ 1 โˆง ๐‘ค ๐‘™ = ๐‘˜ ๐‘™+ โˆ’ ๐‘˜ ๐‘™โˆ’ โˆ€๐‘™ โ‰  ๐‘—, (2.36) and โˆ’ ๐‘ค ๐‘— = ๐‘˜ +๐‘— โˆ’ ๐‘˜ โˆ’๐‘— โˆ“ 1 โˆง โˆ’๐‘ค ๐‘™ = ๐‘˜ ๐‘™+ โˆ’ ๐‘˜ ๐‘™โˆ’ โˆ€๐‘™ โ‰  ๐‘—, (2.37) according to Eq. (2.20). Resonances of order ๐‘š res appear for the first time in the normal form transformation step of order ๐‘š NF = ๐‘š res โˆ’ 1. Consider a four dimensional phase space system (๐‘› = 2) without parameter dependence, where the eigenvalue phases ๐œ‡๐‘– satisfy the following order seven resonance 2๐œ‡1 โˆ’ 5๐œ‡2 = โˆ’4๐œ‹. This   corresponds to the tune resonance condition of โˆ’2๐œˆ1 + 5๐œˆ2 = 2 denoted by ( ๐‘ค, ๐‘‡ ยฎ ๐‘”) = (โˆ’2, 5) , 2 . Given ๐‘ค,ยฎ the corresponding terms S ยฑ ยฎ+ ยฎโˆ’ ยฎ๐œ‚ in the normal form that encounter this resonance ๐‘š( ๐‘˜ , ๐‘˜ , ๐‘˜ ), ๐‘— are determined by all vectors ๐‘˜ and ๐‘˜ยฎ โˆ’ that satisfy the conditions in Eq. (2.36) and Eq. (2.37). ยฎ + Hence, the first terms of the normal form to encounter this resonance are the sixth order complex 18 conjugate terms S+ and S โˆ’ (2.38) 6((0,5)๐‘‡ ,(1,0)๐‘‡ ),1 6((1,0)๐‘‡ ,(0,5)๐‘‡ ),1 as well as S โˆ’ and S + . (2.39) 6((0,4)๐‘‡ ,(2,0)๐‘‡ ),2 6((2,0)๐‘‡ ,(0,4)๐‘‡ ),2 Hence, for each subspace, one complex conjugate pair survives due to the resonance between ๐œ‡1 and ๐œ‡2 , which break the rotational symmetry structure of the resulting normal form. As we will see later on and as discussed in [98], resonances in the tune space correspond to fixed point structures in the phase space, which often yields fascinating behavior especially for low order resonances. 2.4 The Normal Form Defect The volume conserving property of Hamiltonian systems expressed by Liouvilleโ€™s theorem is maintained by the normal form transformation. Given the rotational invariants of the normal form, the size of the phase space volume is determined by the normal form radii. Accordingly, the normal form phase space radii constitute invariants of motion up to the order of the normal form transformation if no resonance conditions were encountered. However, they are usually not invariants of the full (order) motion. While the expansion of the transfer map improves in accuracy with every additional order considered, the same is not guaranteed for the normal form transformation. It is unknown how well or even if the normal form converges with higher orders. This is due to its sensitivity to resonances, which may initiate asymptotic behavior once the order of a close-by resonance is reached. The higher the order of the computation, the more resonances are potentially relevant. Depending on the complexity of the original transfer map, it is usually unpredictable which resonances may affect the normal form and in what way. However, if the normal form transformation converges, its high order limit will yield the exact invariants. In the case of exact invariants, the system is integrable and can be transformed into a trivial system by introducing the invariants as variables. Those variables are known as action-angle coordinates, where the action is constant and unique for each phase space curve and each point on 19 the phase space curve is associated with the action-angle. For complex systems such as the ones discussed in this thesis, there are no exact invariants that can be expressed in terms of finite order terms. Thus, tools to assess the error of the calculated pseudo-invariants in the form of normal form radii are useful. The normal form defect represents the inaccuracy of the normal form radii as invariants and is locally defined for each phase space state. Given an origin preserving fixed point map M (๐‘ž, ๐‘) = (๐‘„, ๐‘ƒ) of a repetitive system and the corresponding normal form transformation A (๐‘ž, ๐‘) = (๐‘ž NF , ๐‘ NF ), the normal form defect ๐‘‘NF (ยฎ๐‘ง0 ) of the phase space state ๐‘งยฎ0 = (๐‘ž, ๐‘) is given by the difference between the normal form radius ๐‘Ÿ (ยฎ๐‘ง1 = M (ยฎ๐‘ง0 )) of the mapped phase space state ๐‘งยฎ1 = M (ยฎ๐‘ง0 ) and the normal form radius ๐‘Ÿ (ยฎ๐‘ง0 ) of the original phase space state ๐‘งยฎ0 . Generally, the normal form radius ๐‘Ÿ of a phase space state ๐‘งยฎ is the magnitude of the vector formed by the normal form phase space state (๐‘ž NF , ๐‘ NF ) = A (ยฎ๐‘ง0 ), specifically โˆš๏ธƒ ๐‘Ÿ (ยฎ๐‘ง) = (๐‘ž NF (ยฎ๐‘ง)) 2 + ( ๐‘ NF (ยฎ๐‘ง)) 2 . (2.40) Accordingly, the normal form defect is given by ๐‘‘NF (ยฎ๐‘ง0 ) = ๐‘Ÿ 1 โˆ’ ๐‘Ÿ 0 = ๐‘Ÿ (ยฎ๐‘ง1 ) โˆ’ ๐‘Ÿ (ยฎ๐‘ง0 ) = ๐‘Ÿ ( M (ยฎ๐‘ง 0 )) โˆ’ ๐‘Ÿ (ยฎ๐‘ง0 ) โˆš๏ธƒ โˆš๏ธƒ = (๐‘ž NF ( M (ยฎ๐‘ง0 ))) + ( ๐‘ NF ( M (ยฎ๐‘ง 0 ))) โˆ’ (๐‘ž NF (ยฎ๐‘ง0 )) 2 + ( ๐‘ NF (ยฎ๐‘ง0 )) 2 . 2 2 (2.41) The application of the one turn map represents the evolution of the system by describing how each phase space state changes after one revolution of the system. The normal form defect indicates how much the normal form radii, i.e., a (pseudo-)invariants of the motion, change between two states of the motion connected by the map M. An increasing normal form radius with time indicates diverging phase space behavior with larger amplitudes, i.e. the normal form defect measures the local rate of divergence per map application. Analyzing the normal form defect for a whole set of states within a certain phase space domain D allows for stability estimations by placing an upper bound on the rate of divergence. The upper bound can be determined in various ways, including rigorous global optimization methods on the normal form defect over the given domain. The upper bound can serve as a Nekhoroshev-type 20 stability estimate [73] that allows for the calculation of the minimum amount of revolutions of the system ๐‘, for which the motion will be guaranteed to stay within the allowed region D: ๐‘Ÿ max โˆ’ ๐‘Ÿ (ยฎ๐‘งini ) ๐‘= with ๐‘งยฎ โˆˆ D, (2.42) max (๐‘‘NF (ยฎ๐‘ง)) where ๐‘Ÿ (ยฎ๐‘ง ini ) is the upper bound of the normal form radius of the initial state of the system and ๐‘Ÿ max is the lower bound of the maximum normal form radius corresponding to motion still within the allowed region D (see Fig. 2.1). ๐‘Ÿ max D ๐‘Ÿ (ยฎ๐‘งini ) ๐‘ ๐‘‘NF (D) Figure 2.1: Schematic illustration of the various normal form quantities involved in the calculation of the minimum iteration number within allowed region D. The concept of a Nekhoroshev-type stability estimate based on the normal form defect is comparable to an augmented Lyapunov function [51]. A regular Lyapunov function ๐ฟ is not increasing along any phase space curve, with ๐ฟ( M (ยฎ๐‘ง )) โ‰ค ๐ฟ(ยฎ๐‘ง ). This works very well for systems with damping. For damped motion in a convex potential, the total energy function can serve as a Lyapunov function. For systems without damping, this is a lot less straightforward. Under the assumption that the normal form algorithm produces a normal form radius which is a true invariant of the motion, the normal form transformation to calculate the normal form radius is a 21 regular Lyapunov function proving eternal stability. However, the errors to the limited floating point accuracy already break this hypothetical scenario. An augmented or pseudo-Lyapunov function ๐ฟโ˜… = ๐ฟ + max (๐‘‘NF (D)) is increasing in a very slow and well estimated way with a verified upper bound on the rate of increase per iteration ๐ฟ ( M (ยฎ๐‘ง)) โ‰ค ๐ฟโ˜… = ๐ฟ (ยฎ๐‘ง) + max (๐‘‘NF (D)) . (2.43) Thus, it cannot prove eternal stability, but it can rigorously estimate the long term stability. See [23] and [48] for a detailed discussion. In [29], this method was successfully used to analyze the long term stability of the Tevatron storage ring at the Fermi National Accelerator Laboratory. However, it can be generally used in dynamical systems applications to assess stability. Particularly, in complex systems where the stability in different phase space regions is not evident, the Nekhoroshev-type stability estimate based on the normal form defect is a great tool to capture the maximum rate of divergence. 2.5 Verified Computations Using Taylor Models (TM) Based on DA vectors (Sec. 2.1), Taylor Models (TM) were developed by Makino and Berz [53, 58, 54, 55, 21, 75] as a structure for rigorously verified computations, which deals much better with issues known from interval arithmetic like the dependency problem [55], the wrapping effect [62, 60, 24], and linear scaling of the overestimation with domain size. Accordingly, the following introduction to TM and their application is largely based on their work [53, 58, 54, 55, 21, 75, 62]. To better understand the advantages of TM, we will first take a quick look at the alternative of using interval arithmetic for verified computations. 2.5.1 Interval Arithmetic Intervals are a basic concept to represent a range of numbers and are often used to capture uncertainty. The interval ๐ผ = [๐‘Ž, ๐‘] = {๐‘ฅ | ๐‘Ž โ‰ค ๐‘ฅ โ‰ค ๐‘} represents all numbers between ๐‘Ž and ๐‘, and the values ๐‘Ž and ๐‘ themselves. 22 The basic interval arithmetic [67, 68, 47] for the addition, subtraction, multiplication, and division of two intervals ๐ผ1 = [๐‘Ž 1 , ๐‘ 1 ] and ๐ผ2 = [๐‘Ž 2 , ๐‘ 2 ] are given by the following operations. The addition yields ๐ผ1 + ๐ผ2 = [๐‘Ž 1 + ๐‘Ž 2 , ๐‘ 1 + ๐‘ 2 ] . (2.44) The subtraction operation ๐ผ1 โˆ’ ๐ผ2 works equivalently by performing the addition of ๐ผ1 with โˆ’๐ผ2 = [โˆ’๐‘ 2 , โˆ’๐‘Ž 2 ]. The multiplication yields ๐ผ1 ยท ๐ผ2 = [min (๐‘Ž 1 ๐‘Ž 2 , ๐‘Ž 1 ๐‘ 2 , ๐‘ 1 ๐‘Ž 2 , ๐‘ 1 ๐‘ 2 ) , max (๐‘Ž 1 ๐‘Ž 2 , ๐‘Ž 1 ๐‘ 2 , ๐‘ 1 ๐‘Ž 2 , ๐‘ 1 ๐‘ 2 )] . (2.45) The division is only possible if the divisor interval does not contain zero. If the divisor does not contain zero, the division ๐ผ1 /๐ผ2 is equivalently defined by multiplying ๐ผ1 with   1 1 1 = , for 0 โˆ‰ ๐ผ2 . (2.46) ๐ผ2 ๐‘2 ๐‘Ž2 This arithmetic provides the mathematically tightest bounds for certain operations like the square of an interval, and when the quantities represented by ๐ผ1 and ๐ผ2 are independent. However, in many cases, the calculated bounds are an overestimation due to the dependency problem, which is easily illustrated by considering the difference between an interval and itself. The result of the expression ๐‘ฅ โˆ’ ๐‘ฅ should be zero, but from the arithmetic above the difference between two identical intervals is ๐ผ โˆ’ ๐ผ = [๐‘Ž, ๐‘] โˆ’ [๐‘Ž, ๐‘] = [โˆ’ (๐‘ โˆ’ ๐‘Ž) , (๐‘ โˆ’ ๐‘Ž)] , (2.47) which has a width of 2 (๐‘ โˆ’ ๐‘Ž) instead of zero width. Compared to DA vectors (see Sec. 2.1), which form a ring structure, intervals do not even form a group structure, because neither for addition nor multiplication there is an inverse for intervals of nonzero width. For the interval evaluation of functions, further rules can be established. Monotonically increasing functions ๐‘“mon% like exp(๐‘ฅ) can be evaluated by   ๐‘“mon% ( [๐‘Ž, ๐‘]) = ๐‘“mon% (๐‘Ž), ๐‘“mon% (๐‘) . (2.48) 23 Monotonically decreasing functions ๐‘“mon& can be equivalently evaluated by   ๐‘“mon& ( [๐‘Ž, ๐‘]) = ๐‘“mon& (๐‘), ๐‘“mon& (๐‘Ž) . (2.49) Trigonometric functions are compositions of monotonically increasing and monotonically decreasing sections, which are well known. Accordingly, the interval evaluation of a trigonometric function can be implemented based on many subcases depending on the size and position of the interval. Considering the function ๐‘“ (๐‘ฅ) = sin(๐œ‹๐‘ฅ/2) โˆ’ exp(๐‘ฅ) and evaluating it over the domain interval ๐ผ1 = [โˆ’1, 1] yields   ๐œ‹๐ผ1 ๐‘“ (๐ผ1 ) = sin โˆ’ exp (๐ผ1 ) = ๐ผ1 โˆ’ [exp (โˆ’1) , exp (1)] (2.50) 2 h i = โˆ’1 โˆ’ ๐‘’, 1 โˆ’ ๐‘’ โˆ’1 โŠ‚ [โˆ’3.718282, 0.632121] . (2.51) We will compare this interval evaluation to the performance of different order Taylor Models in the following section. 2.5.2 Taylor Models Taylor Models [53, 58, 54, 55, 21, 75] are remainder-enhanced DA vectors. The DA part of the TM of the function ๐‘“ is an ๐‘šth order Taylor polynomial in form of a regular DA vector. The remainder part complements this by rigorously verified bounds on the error of using the truncated Taylor expansion of ๐‘“ up to order ๐‘š in form of a DA vector compared to ๐‘“ itself. In contrast to regular DA vectors, the function ๐‘“ must be (๐‘š + 1) times continuously partially differentiable to evaluate the reminder using the Taylor Remainder Theorem. Additionally, TM need to be defined over a domain D to be able to rigorously bound the remainder. h i The Taylor Remainder Theorem says: Given a function ๐‘“ : D ยฎ = ๐‘Ž, ยฎ ๐‘ยฎ โŠ‚ R๐‘› โ†’ G โŠ‚ R being (๐‘š + 1) times continuously partially differentiable on the domain D ยฎ with ๐‘ฅยฎ0 โˆˆ D.ยฎ Then for each 24 ยฎ there is an ๐œ‚ โˆˆ (0, 1) such that ๐‘ฅยฎ โˆˆ D  ๐‘˜   ๐‘š+1 ๐‘š ยฎ ( ๐‘ฅยฎ โˆ’ ๐‘ฅยฎ0 ) ยท โˆ‡ ๐‘“ ( ๐‘ฆยฎ) ยฎ ( ๐‘ฅยฎ โˆ’ ๐‘ฅยฎ0 ) ยท โˆ‡ ๐‘“ ( ๐‘ฆยฎ) โˆ‘๏ธ ๐‘ฆยฎ ๐‘ฆยฎ ๐‘“ ( ๐‘ฅยฎ) = + , (2.52) ๐‘˜! (๐‘š + 1)! ๐‘˜=0 ๐‘ฆยฎ=ยฎ ๐‘ฅ0 ๐‘ฅ 0 + ( ๐‘ฅยฎโˆ’ยฎ ๐‘ฆยฎ=ยฎ ๐‘ฅ0 ) ๐œ‚ | {z } | {z } P๐‘š, ๐‘“ E๐‘š,D, ๐‘“ where P๐‘š, ๐‘“ is the polynomial part and E is an expression for the remainder. A Taylor Model is characterized by its order ๐‘š, the function ๐‘“ that it is representing, and the domain D over which the representation of ๐‘“ is within the verified bounds of the Taylor Model. We denote a Taylor Model with   T๐‘š,D, ๐‘“ = P๐‘š, ๐‘“ , ๐œ– ๐‘š,D, ๐‘“ , (2.53) where P๐‘š, ๐‘“ is the Taylor polynomial term of order ๐‘š and ๐œ– ๐‘š,D, ๐‘“ is a rigorous verified estimation ๐œ– ๐‘š,D, ๐‘“ of the remainder size over the domain D such that for function ๐‘“ ๐‘ฅ ) โˆ’ P๐‘š, ๐‘“ (ยฎ ๐‘“ (ยฎ ๐‘ฅ ) โ‰ค ๐œ– ๐‘š,D, ยฎ ๐‘“ โˆ€ยฎ ยฎ ๐‘ฅ โˆˆ D. (2.54) A Taylor Model can be visualized as a tube that wraps around the ๐‘šth order DA representation with a distance ๐œ– such that the original expression is guaranteed to lie within the tube over the given domain D (see Fig. 2.2). Except for order ๐‘š = 1, the Taylor Model bounding of ๐‘“ significantly outperforms the interval bounding. The tightness of the bounding also improves drastically with higher order Taylor Models. With every additional order, the polynomial part clings closer to ๐‘“ , and the reminder gets smaller and smaller. This tighter and tighter bounding with higher orders shows how the DA part of the Taylor Models avoids more and more of the dependency problem. Dependent expressions like 1 + ๐‘ฅ โˆ’ ๐‘ฅ, which may arise as the first order part of expressions like exp(๐‘ฅ) โˆ’ sin(๐‘ฅ) are reduced to just 1 + 0 in the DA part of the Taylor Model description. As we saw in Sec. 2.5.1, interval arithmetic is not able to avoid this dependency problem. 25 2 Order ๐‘š = 1 Order ๐‘š = 2 1 0 ๐‘ฆ -1 -2 -3 -4 -0.6 Order ๐‘š = 3 Order ๐‘š = 4 -0.8 -1 -1.2 ๐‘ฆ -1.4 -1.6 -1.8 -2 -0.9 Order ๐‘š = 5 Order ๐‘š = 6 -1 -1.1 -1.2 ๐‘ฆ -1.3 -1.4 ๐‘“ (๐‘ฅ) -1.5 P๐‘š, ๐‘“ -1.6 P๐‘š, ๐‘“ ยฑ ๐œ– ๐‘š,D, ๐‘“ -1.7 ๐‘“ (D) -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 ๐‘ฅ ๐‘ฅ Figure 2.2: Verified representation of ๐‘“ (๐‘ฅ) = sin(๐œ‹๐‘ฅ/2) โˆ’ exp(๐‘ฅ) over the domain D = ๐ผ1 = [โˆ’1, 1] with interval methods using ๐‘“ (D) and with Taylor Models ( P๐‘š, ๐‘“ , ๐œ– ๐‘š,D, ๐‘“ ) of various orders ๐‘š. The original function ๐‘“ (๐‘ฅ) is indicated by the black line, while its DA polynomial representation is shown in green. The bounds at a distance ๐œ– ๐‘š,D, ๐‘“ from the DA polynomial are red. The two straight blue lines indicate the bounds of the interval evaluation. Note that the scale of the ๐‘ฆ axis is changing to better illustrate the tightness of the Taylor Model representation with higher orders. Accordingly, the interval bounds are only shown for order ๐‘š = 1 and order ๐‘š = 2. The fourth order Taylor Model representation of the function ๐‘“ (๐‘ฅ) = sin(๐œ‹๐‘ฅ/2) โˆ’ exp(๐‘ฅ) over the domain D = ๐ผ1 = [โˆ’1, 1] would be ! (๐œ‹ โˆ’ 2)๐‘ฅ ๐‘ฅ 2 (๐œ‹ 3 โˆ’ 8)๐‘ฅ 3 ๐‘ฅ 4 T4,๐ผ1 , ๐‘“ (๐‘ฅ) = โˆ’1 + โˆ’ โˆ’ โˆ’ , 0.102345 . (2.55) 2 2 48 24 26 2.6 Taylor Model based Verified Global Optimizers The goal of a verified global optimizer [12, 69, 29, 63, 57, 43] is finding the optimum of a given scalar objective function ๐‘“ ( ๐‘ฅยฎ) of ๐‘›var variables ๐‘ฅ๐‘– over a predefined ๐‘›var dimensional global search domain box B. ยฎ Without loss of generality, it is assumed that the optimum is a minimum. If the optimum is a maximum, consider the optimization of โˆ’ ๐‘“ ( ๐‘ฅยฎ). Ideally, the result of global optimization yields the minimum ๐‘“ โ˜… of the objective function ๐‘“ ( ๐‘ฅยฎ) and all locations ๐‘ฅยฎโ˜…, where the minimum is assumed within the global search domain box B. ยฎ However, straightforward and exact analytic solutions of the optimization problem only exist for elementary objective functions. As soon as higher order terms and multiple variables are involved, iterative algorithms to track down the optimum are inevitable. Consequently, results are often only approximations of the actual minimum and all their locations where it is assumed. Verified global optimizers compensate for the shortcoming of being unable to pinpoint the exact minimum by yielding rigorously verified bounds on the minimum and its locations. The fundamental idea of a verified global optimization algorithm is the efficient elimination of subdomains/subboxes of the initial search box B ยฎ by proving that those eliminated subboxes do not contain the minimum. The basic steps of the algorithm are the following: 1. Split domain box B ยฎ into subdomains Bยฎ๐‘–. 2. Determine a lower bound ๐‘“๐‘–,LB of ๐‘“ over ๐‘ฅยฎ โˆˆ B ยฎ๐‘–. 3. Calculate/Update the cutoff value C โ€“ the currently lowest known upper bound of the minimum. 4. Eliminate all boxes B๐‘– with a lower bound ๐‘“๐‘–,LB larger than the cutoff value C . 5. Restart the algorithm at step 1 for each of the non-eliminated domain boxes B ยฎ #. ๐‘– The more subdomain boxes are eliminated in step 4 in each iteration, the more effective the algorithm. Accordingly, it is essential to use methods for very tight bounding in step 2 (making ๐‘“๐‘–,LB as large as possible), and to use heuristics to significantly improve the cutoff value C in step 3, making it as small as possible. 27 For the determination of the cutoff value C in step 3, any method or combination of methods that produce a tight verified upper bound on the global minimum of the search domain are useful. A typical technique is the verified evaluation of individual points within the domain box. The testing points are chosen either randomly in a Monte-Carlo based approach or by heuristics, e.g., the results of non-verified optimization over the domain. Depending on the computational effort of those methods, the improvement of the cutoff value and its benefits for the algorithm must be weighed against the computation time of the cutoff method. For step 2, Taylor Models (see Sec. 2.5) are particularly useful, especially high order Taylor Models, since they allow for very tight bounding compared to interval methods. This property can mainly be ascribed to the avoidance of the dependency problem due to the DA vector part of the TM. For very complex objective functions like the normal form defect (see Sec. 2.4), the evaluation with very high order Taylor Models (e.g. order ten) can take considerably more time compared to evaluations with lower order Taylor Models (e.g. order three). Again, the benefits of the more precise bounding with high order Taylor Model evaluation have to be weighed against the associated computation time. A rule of thumb is that the larger the evaluation domain and the more complex the objective function, the larger the benefit of higher order Taylor Models. For the rigorous bounding of Taylor Models, there are multiple approaches. The standard method uses order bounds, where the terms belonging to each order are bound and summed up together with the remainder bound. More sophisticated methods are discussed in great detail in [64]. They can be briefly summarized as follows. The linear dominated bounder (LDB) is very efficient for linear dominated domains. The quadratic dominated bounder (QDB) is good at determining the minimum of a multidimensional quadratic dominated function but losses its efficiency with very high dimensional problems. The quadratic fast bounder (QFB) is not as exact as the QDB but very efficient in providing a good lower bound near a local minimum, where the Hessian matrix of the objective function over the domain is positive definite. To avoid an infinite continuation of the splitting, stop conditions are implemented, which are checked before a domain box is split. A typical stop condition sets a lower bound on the size of 28 the domain, either by setting a lower bound on the volume of the domain box or its side length. Non-eliminated domain boxes below such a threshold values are not split. Another possible stop condition is a lower bound on the tightness of the bounding of the minimum of the objective function rather than the domain size. With such a stop condition in place, the algorithm would not split a non-eliminated domain box over which the bounds of the minimum are tighter than a certain given value. This is particularly useful if the exact minimum is not relevant but rather the order of magnitude of the minimum. 29 CHAPTER 3 AN EXAMPLE-DRIVEN WALK-THROUGH OF THE DA NORMAL FORM ALGORITHM This chapter is based on my arXiv preprint and MSU Report MSUHEP-190617 Introduction to the Differential Algebra Normal Form Algorithm using the Centrifugal Governor as an Example [94]. We provide a very detailed description of the steps involved in the DA normal form algorithm (Sec. 2.3) and their implications for the normal form using the example of the centrifugal governor. We pick this example because it is one dimensional and the derivation of the equations of motion and the linearization of the motion are well known. This understanding yields the groundwork for the non-trivial analysis of the nonlinear phenomena using the steps of the DA normal form algorithm. 3.1 The Centrifugal Governor The centrifugal governor (see Fig. 3.1) is a device involving gravitational and centrifugal forces with the rotation axis parallel to the direction of the gravitational force. We consider a mathematically idealized governor, which consists of two massless rods of equal length ๐‘… suspended in a common Figure 3.1: Schematic illustration of centrifugal governor. 30 plane with the rotation axis. A point mass ๐‘š is attached at the end (opposite to where the rod is mounted) of each of the rods. The angle between the rotation axis and the rod is denoted by the angle ๐œ™. A mechanism links the two rods and the rotation axis, which guarantees identical angles and therefore identical behavior on both sides. An external torque applied via the rotation axis ensures that the rotation frequency ๐œ” of the centrifugal governor arms is kept constant. In the usual application of a centrifugal governor, the rotation frequency is not fixed but negatively coupled to the angle ๐œ™ through an additional mechanism external to the governor itself. This additional mechanism makes the system self regulating by decreasing ๐œ” for an increase in ๐œ™. Accordingly, in those applications, e.g. the steam engine, the rotation frequency ๐œ” changes during the regulating process. However, as already mentioned above, for the introduction to the DA normal form algorithm, we consider the motion of the system for a fixed rotation frequency ๐œ”, i.e. no self-regulating coupling mechanism between ๐œ™ and ๐œ”. 3.1.1 Units To limit the number of parameters in the following calculations to just the rotation frequency ๐œ”, we scale time, distance, and mass in such a way that the mass ๐‘š, the gravitational constant ๐‘”, and the length of the rods ๐‘… are all equal to one in their respective scaled units and therefore disappear from the equations. Specifically, mass is considered in units of the point mass ๐‘š, distances are considered in units of the rod length ๐‘…, and time is considered in units of v u t ๐‘…[m] ๐‘‡0 [s] = h i, (3.1) ๐‘” m2 s such that the gravitational constant ๐‘” equal one in units of distance ๐‘… and time ๐‘‡0 . 3.1.2 The Equilibrium Point For any given fixed rotation frequency ๐œ”, there is an angle ๐œ™0 so that ๐œ™(๐‘ก) = ๐œ™0 is a solution of the motion of the centrifugal governor arms. This equilibrium angle is characterized by the alignment of the rods with the vector sum of the vertical gravitational force ๐นgrav and the radial centrifugal 31 force ๐นcent such that there is no torque acting on the rods in the common plane of the rods and the rotation axis. For any frequency ๐œ”, ๐œ™0 = 0 satisfies this requirement, since the centrifugal force is zero and there is only the gravitational force acting vertically downwards. However, if the rotation frequency ๐œ” is sufficiently high enough (see Eq. (3.3)), a bifurcation of the equilibrium angle occurs โ€“ the angle ๐œ™0 = 0 becomes an unstable equilibrium state, while stable equilibrium angle ๐œ™0 (๐œ”) > 0 arises, which satisfies the alignment condition with ๐นcent ๐‘š๐œ”2 ๐‘… sin ๐œ™0 tan ๐œ™0 = = = ๐œ”2 sin ๐œ™0 . (3.2) ๐นgrav ๐‘š๐‘” For ๐œ™0 > 0, this corresponds to   1 1 cos ๐œ™0 = โ‡’ ๐œ™0 = arccos for ๐œ” > 1 = ๐œ”min . (3.3) ๐œ”2 ๐œ”2 Fig. 3.2 visualizes the stable equilibrium angle as a function of the rotation frequency ๐œ”. Since the vertical contribution of the gravitational force to the vector sum is nonzero and independent of the rotation frequency, an equilibrium angle of ๐œ™0 = 90ยฐ is only approached asymptotically for the rotation frequency ๐œ” approaching infinity. The bifurcation of the equilibrium state at ๐œ”min = 1 is also clearly visible. 90 80 equilibium angle ๐œ™0 [deg] 70 60 50 40 30 20 arccos(1/๐œ”2 ) for ๐œ”>1=๐œ”min n 10 0 for ๐œ”โ‰ค1=๐œ”min 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 ๐œ” Figure 3.2: Illustration of the stable equilibrium angle ๐œ™0 of the arms of the centrifugal governor as a function of the rotation frequency ๐œ”. For ๐œ” > ๐œ”min = 1, ๐œ™0 = 0 is an unstable equilibrium angle. 32 Tab. 3.1 lists stable equilibrium angles for some specific rotation frequencies, especially for the fast-changing region between ๐œ” = 1 and ๐œ” = 2. Table 3.1: List of stable equilibrium angles ๐œ™0 of the centrifugal governor arms for some specific rotation frequencies ๐œ”. ๐œ” ๐œ™0 [deg] ๐œ™0 [rad] 1 0ยฐ 0 โˆš โˆš4 2/ 3 30ยฐ ๐œ‹/6 โˆš 4 2 45ยฐ ๐œ‹/4 โˆš 2 60ยฐ ๐œ‹/3 2 โ‰ˆ 75.52ยฐ โ‰ˆ 1.318 20 โ‰ˆ 89.86ยฐ โ‰ˆ 1.568 lim๐œ”โ†’โˆž 90ยฐ ๐œ‹/2 3.1.3 The Equations of Motion To understand the dynamics of the centrifugal governor arms around an equilibrium state, we derive the equations of motion for one of the two masses starting with the Lagrangian formulation of the problem. It yields ! ๐‘š  ยค2 2  ๐œ™ยค2 โˆ’๐œ” 2 sin2 ๐œ™ ๐ฟ= ๐œ™ ๐‘… + ๐œ”2 ๐‘… 2 sin2 ๐œ™ โˆ’ ๐‘š๐‘”๐‘… (1 โˆ’ cos ๐œ™) = โˆ’ + (1 โˆ’ cos ๐œ™) , (3.4) 2 2 2 | {z } ๐‘ˆeff where ๐‘ˆeff is the effective or centrifugal-gravitational potential. In Fig. 3.3, we illustrate the centrifugal-gravitational potential ๐‘ˆeff for multiple rotation frequencies ๐œ”. The minimum of the effective potential well corresponds to the stable equilibrium angle discussed in Sec. 3.1.2. The axis notations indicate that the width and the depth of the potential increase with increasing rotation frequency ๐œ”. The higher the rotation frequency ๐œ”, the less relevant are the gravitational influences and the deeper and the more symmetric the potential well. The asymmetry of the effective potential is also apparent in the dynamics of the system, which we discuss in Sec. 3.1.4. 33 0 โˆš โˆš 4 0 โˆš 0 0 ๐œ” = 2/ 3 ๐œ”= 2 ๐œ”=2 ๐œ” = 20 ๐œ™0 = 30ยฐ ๐œ™0 = 60ยฐ ๐œ™0 โ‰ˆ 75.52ยฐ ๐œ™0 โ‰ˆ 89.86ยฐ ๐‘ˆeff ๐‘ˆeff ๐‘ˆeff ๐‘ˆeff โˆ’150 โˆ’0.2 โˆ’0.01 โˆ’1 0 15 30 45 0 30 60 90 0 30 60 90 120 0 60 120 180 ๐œ™ [deg] ๐œ™ [deg] ๐œ™ [deg] ๐œ™ [deg] Figure 3.3: Potential well of ๐‘ˆeff for multiple oscillation frequencies ๐œ”. The equilibrium angle ๐œ™0 corresponds to the minimum of the potential well. โˆš For the rest of the chapter, we will focus on the case ๐œ” = 2, which yields a clear 2:1 asymmetry left and right of its equilibrium angle. To continue the derivation of the equations of motion, we derive the generalized canonical momentum ๐‘ ๐œ™ to the position variable ๐œ™ from the Lagrangian, where ๐‘‘๐ฟ ๐‘๐œ™ = = ๐‘š๐‘… 2 ๐œ™ยค = ๐œ™. ยค (3.5) ๐‘‘๐œ™ยค Using the Legendre transformation, the Hamiltonian ๐‘ 2๐œ™๐‘š๐œ”2 ๐‘… 2 sin2 ๐œ™ ๐‘ 2๐œ™ ๐ป= โˆ’ + ๐‘š๐‘”๐‘… (1 โˆ’ cos ๐œ™) = + ๐‘ˆeff = ๐ธ (3.6) 2๐‘š๐‘… 2 2 2 is obtained, which is not explicitly time dependent and therefore a constant of motion. The Hamiltonian also happens to correspond to the energy ๐ธ of this system. The equations of motions are derived from the Hamiltonian via Hamiltonโ€™s equations where ๐‘‘๐ป ๐‘๐œ™ ๐œ™ยค = = = ๐‘๐œ™ (3.7) ๐‘‘๐‘ ๐œ™ ๐‘š๐‘… 2 ๐‘‘๐ป   and ๐‘ยค ๐œ™ = โˆ’ = โˆ’๐‘š๐‘”๐‘… sin ๐œ™ + ๐‘š๐œ”2 ๐‘… 2 sin ๐œ™ cos ๐œ™ = sin ๐œ™ ๐œ”2 cos ๐œ™ โˆ’ 1 . (3.8) ๐‘‘๐œ™ In coordinates (๐›ฟ๐œ™, ๐›ฟ ๐‘ ๐œ™ ) relative to the equilibrium state (๐œ™0 , 0), the equations of motions are d๐›ฟ๐œ™ d๐›ฟ ๐‘ ๐œ™   = ๐›ฟ ๐‘๐œ™ and = sin (๐œ™0 + ๐›ฟ๐œ™) ๐œ”2 cos (๐œ™0 + ๐›ฟ๐œ™) โˆ’ 1 . (3.9) d๐‘ก d๐‘ก 34 3.1.4 Illustration of System Dynamics With the equations of motion relative to the equilibrium state (Eq. (3.9)) and the understanding of how the shape of the effective potential well changes with the rotation frequency ๐œ”, we can now interpret the dynamics of the centrifugal governor when the angle of the rods is perturbed from the equilibrium angle ๐œ™0 (๐œ”). โˆš In Fig. 3.4, the dynamics of the rods are shown for a rotation frequency of ๐œ” = 2, which corresponds to an equilibrium angle of ๐œ™0 = 60ยฐ. While the oscillation is periodic, it is asymmetric โˆš around the equilibrium point, as we would expect from the asymmetric effective potential for ๐œ” = 2 in Fig. 3.3. The asymmetry of the oscillation is larger, the larger the angle during initiation. The maximum downward angle displacement (often more generally referred to as amplitude) and the maximum upward angle displacement of the governorโ€™s arms relative to their equilibrium angle are related through the effective potential, which corresponds to the energy of the vertical motion for 90 80 angle ๐œ™ of governor arms [deg] 70 60 50 40 30 20 โˆš 10 ๐œ”= 2 0 0 2 4 6 8 10 12 โˆ’20 140 16 โˆ’40 20 40 time [๐‘‡0 ] ๐œ™ยค [deg/๐‘‡0 ] โˆš Figure 3.4: Dynamics of the centrifugal governor for a rotation frequency of ๐œ” = 2. The centrifugal governor arms were initiated with ๐œ™ยค = ๐‘ ๐œ™ = 0 and at the following angles: 60ยฐ, 65.5ยฐ, 69.5ยฐ, 73.5ยฐ, 77.5ยฐ, 81.5ยฐ, 85.5ยฐ, and 89.5ยฐ. The left plot shows the oscillatory behavior around the equilibrium angle at ๐œ™0 = 60ยฐ over time. The right plot shows the stroboscopic phase space behavior from repetitive map evaluation. To relate the phase space behavior to the position behavior in time, the ๐œ™ axis of both plots are aligned. 35 ๐›ฟ ๐‘ ๐œ™ = 0. For both those angle displacements, the effective potential has the same maximum value or โ€˜invariant amplitudeโ€™ corresponding to the energy. The maximum amplitudes in the momentum space in the right plot of Fig. 3.4 are related the same way. In other words, the phase space motion in Fig. 3.4 corresponds to contour lines of the energy. For future reference, it is useful to associate the term โ€˜amplitudeโ€™ not only with a physical displacement or a maximum/minimum momentum but also with an abstract quantity that relates all the different versions of phase space amplitudes like the energy in this case. Apart from the asymmetric upward and downward position amplitudes, the left plot in Fig. 3.4 clearly shows a change in the period of oscillation depending on the angle during initiation, or more generally speaking, depending on the invariant amplitude of the motion, e.g., the total energy of the system. The larger the amplitude, the longer is the period of oscillation. This is particularly prominent for the oscillation with the largest amplitude. It is also obvious, especially for the larger amplitudes that the relation between the amplitude and the period is nonlinear. However, there is no trivial way of extracting this nonlinear relation between the amplitude and the period of oscillation from the equations of motion and/or the energy. Additionally, if we were unaware of the function for the effective potential and energy, or were considering a more complex system, it would also be very difficult to relate the different phase space amplitudes to each other. The DA normal form algorithm generates both relations in an automated process up to calculation order. In the order-by-order process, it determines an invariant amplitude up to calculation order as a function of the original phase space variables and also determines the period of oscillation as a function of that invariant amplitude. All the normal form algorithm requires is an origin preserving transfer map (see Sec. 2.2), which represents the flow of the ODEs (see Eq. (3.9)) relative to the linearly stable fixed point of the considered phase space motion. For the centrifugal governor example, the equilibrium phase space state (๐œ™0 , 0) constitutes such a phase space fixed point, as the right plot in Fig. 3.4 already indicated. In other words, we require a functional description of how the relative phase space state ๐‘งfin = (๐›ฟ๐œ™fin , ๐›ฟ ๐‘ ๐œ™,fin ) after a fixed time ๐‘ก0 depends on the initial relative phase space 36 state ๐‘งini = (๐›ฟ๐œ™ini , ๐›ฟ ๐‘ ๐œ™,ini ). DA based maps (Sec. 2.2) can provide this functional description up to arbitrary order. We will use them to represent the dynamics around the equilibrium state โˆš corresponding to a rotation frequency of ๐œ” = 2, for the later analysis with the DA normal form algorithm. 3.2 Map Calculation via Integration As mentioned above, the following analysis of the centrifugal governor considers the system at a โˆš fixed rotation frequency of ๐œ” = 2. We are interested in the dynamics around the corresponding equilibrium state of the centrifugal governor arms at (60ยฐ, 0). The goal of this section is to generate a DA map describing the phase space dynamics relative to that equilibrium state. For consistency with the notation introduced in Sec. 2.3, we denote the phase space coordinates  relative to the equilibrium point with (๐‘ž 0 , ๐‘ 0 ) instead of the previously used ๐›ฟ๐œ™, ๐›ฟ ๐‘ ๐œ™ . We will also conduct the calculations in radians rather than degrees due to their slightly easier implementation. The map is calculated by integrating the ODEs (see Eq. (3.9)) from the initial phase space state   โˆš   ๐œ‹  (๐‘ž ini , ๐‘ ini ) = ๐œ™0 ๐œ” = 2 + ๐›ฟ๐œ™, ๐›ฟ ๐‘ ๐œ™ = + ๐‘ž0, ๐‘0 (3.10) 3 from ๐‘ก = 0 until ๐‘ก = ๐‘ก0 = 1. Since the flow of the ODEs in Eq. (3.9) remains expanded around the equilibrium state for any ๐‘ก0 , the time of the integration can be chosen freely. The resulting map of the integration M0 = (๐‘„(๐‘ž 0 , ๐‘ 0 ), ๐‘ƒ(๐‘ž 0 , ๐‘ 0 ))๐‘‡ has the following form: M0 = C + L + ๐‘š U๐‘š , where the constant part is denoted by C , the linear part with L and each of ร the nonlinear parts of order ๐‘š with U๐‘š . Since the system is expanded around the equilibrium point, the constant part of the map corresponds to the equilibrium state (๐œ‹/3, 0). The following explicit 37 formulation of M0 up to order three introduces the notation of various coefficients of the map: ยฉ M+ (๐‘ž 0 , ๐‘ 0 ) ยช ยฉ๐‘„ (๐‘ž 0 , ๐‘ 0 ) ยช ยฉ ๐‘ž const ยช ยฉ (๐‘„|๐‘ž 0 ) (๐‘„| ๐‘ 0 ) ยช ยฉ ๐‘ž 0 ยช M0 (๐‘ž 0 , ๐‘ 0 ) = ยญยญ 0 ยฎ=ยญ ยฎ=ยญ ยฎ+ยญ ยฎยญ ยฎ โˆ’ ยฎ ยญ ยฎ ยญ ยฎ ยญ ยฎยญ ยฎ M0 (๐‘ž 0 , ๐‘ 0 ) ๐‘ƒ (๐‘ž 0 , ๐‘ 0 ) ๐‘ const (๐‘ƒ|๐‘ž 0 ) (๐‘ƒ| ๐‘ 0 ) ๐‘ 0 ยซ ยฌ ยซ ยฌ ยซ ยฌ ยซ ยฌยซ ยฌ | {z } | {z } C L + ยฉU2(2,0) ยช 2 ยฉU2(1,1)+ ยช + ยฉU2(0,2) ยช 2 +ยญยญ ยฎ ๐‘ž ยฎ 0 ยญ โˆ’+ ยญ ๐‘ž ๐‘ ยฎ 0 0 ยญ โˆ’ ยฎ + ยญ ยฎ๐‘ ยฎ 0 โˆ’ U2(2,0) U2(1,1) U2(0,2) ยซ ยฌ ยซ ยฌ ยซ ยฌ | {z } U2 + ยฉU3(3,0) ยช 3 ยฉU3(2,1)+ + ยฉU3(1,2) + ยฉU3(0,3) ยช 3 2 ยช 2 ยช +ยญยญ ยฎ ๐‘ž + ยญ ยฎ ๐‘ž ๐‘ 0 + ยญ ยฎ ๐‘ž 0 ๐‘ + ยญ ยฎ ๐‘ +... (3.11) โˆ’ ยฎ 0 ยญ โˆ’ ยฎ 0 ยญ โˆ’ ยฎ 0 ยญ โˆ’ ยฎ 0 U3(3,0) U3(2,1) U3(1,2) U3(0,3) ยซ ยฌ ยซ ยฌ ยซ ยฌ ยซ ยฌ | {z } U3 The position ๐‘„ and momentum ๐‘ƒ components of the map M0 correspond to the upper and lower component and are denoted by โ€˜+โ€™ and โ€˜-โ€™, respectively. The coefficients in the upper and lower component for the nonlinear ๐‘š(= ๐‘Ž + ๐‘)th order terms ๐‘ž ๐‘Ž ๐‘ ๐‘ are denoted by U๐‘š(๐‘Ž,๐‘) ยฑ . The coefficients in the linear matrix (๐‘Ž|๐‘) indicate the factor with which ๐‘Ž is linearly dependent on ๐‘. The following Tab. 3.2 lists the values of the coefficients in Eq. (3.11) above. The integration was performed with an order 20 Picard-iteration based integrator with stepsize โ„Ž = 10โˆ’3 over 1000 iterations within COSY INFINITY. Details on the implementation of the integrator are given in [93]. 3.3 The DA Normal Form Algorithm In Sec. 2.3, the general DA normal form algorithm [19] was introduced for a linearly stable 2๐‘› dimensional system with optional parameter dependence. This section provides a detailed example- driven walk-through of the differential algebra based normal form algorithm for the symplectic one dimensional (1D) system of the centrifugal governor without a parameter dependence. The normal form resulting from the DA normal form algorithm constitutes circular motion with a quasi-invariant as radius and only normal form phase space amplitude (and parameter) dependent angle advancements. Fig. 3.5 illustrates the oscillatory phase space behavior of the governorโ€™s arms around the equilibrium point (left plot already seen in different orientation in Fig. 3.4) and compares 38 โˆš Table 3.2: Integration result for map around equilibrium state (๐œ™0 (๐œ” = 2) = ๐œ‹/3, 0) integrated until ๐‘ก = 1 using an order 20 Picard-iteration based integrator with stepsize โ„Ž = 10โˆ’3 over 1000 iterations within COSY INFINITY. The component M+0 = ๐‘„(๐‘ž 0 , ๐‘ 0 ) is on the left, Mโˆ’ 0 = ๐‘ƒ(๐‘ž 0 , ๐‘ 0 ) on the right. Order Coeff. Value Coeff. Value 0 ๐‘ž const 1.04719755 ๐‘ const 0 1 (๐‘„|๐‘ž 0 ) 0.33918599 (๐‘ƒ|๐‘ž 0 ) -1.15214118 1 (๐‘„| ๐‘ 0 ) 0.76809412 (๐‘ƒ| ๐‘ 0 ) 0.33918599 2 + U2(2,0) -0.44622446 โˆ’ U2(2,0) -0.55821731 2 + U2(1,1) -0.29304415 โˆ’ U2(1,1) -0.64033440 2 + U2(0,2) -0.08403817 โˆ’ U2(0,2) -0.29304415 3 + U3(3,0) 0.31844278 โˆ’ U3(3,0) 0.50817317 3 + U3(2,1) 0.29904862 โˆ’ U3(2,1) 0.76091921 3 + U3(1,2) 0.13758223 โˆ’ U3(1,2) 0.46230241 3 + U3(0,3) 0.03017663 โˆ’ U3(0,3) 0.13758223 it to its associated rotationally invariant phase space behavior in the normal form representation. The orientation of the phase space in Fig. 3.5 is according to the usual convention, where the position ๐‘ž is on the horizontal axis and the momentum ๐‘ on the vertical axis. In Fig. 3.4, this convention a) b) Figure 3.5:โˆš Phase space behavior of the centrifugal governor arms around their equilibrium angle of ๐œ™0 (๐œ” = 2) = 60ยฐ provided by a tenth order Poincarรฉ map of the system. a) shows the original phase space behavior. b) shows the associated circular behavior in normal form. 39 was ignored for the sake of a better understanding when comparing the phase space behavior to the position behavior over time. Accordingly, the asymmetry with larger downwards amplitudes is shown in the horizontal (๐œ™) direction in Fig. 3.5a. The transformation steps of the normal form algorithm are done order by order. With each transformation step, the index of the map and the variables is going to increase by 1, i.e. as a result of the first (order) transformation we get M1 dependent on the variables (๐‘ž 1 , ๐‘ 1 ). For each order ๐‘š there is a transformation A๐‘š and its inverse Aโˆ’1 ๐‘š , which are applied to resulting map of the previous transformation M๐‘šโˆ’1 to yield the resulting map of the ๐‘šth order transformation M๐‘š (๐‘ž ๐‘š , ๐‘ ๐‘š ) = ( A๐‘š โ—ฆ M๐‘šโˆ’1 โ—ฆ Aโˆ’1 ๐‘š )(๐‘ž ๐‘š , ๐‘ ๐‘š ). (3.12) The transformation Aโˆ’1 ๐‘š transforms (๐‘ž ๐‘š , ๐‘ ๐‘š ) to (๐‘ž ๐‘šโˆ’1 , ๐‘ ๐‘šโˆ’1 ), which are the variables of the map of the previous order M๐‘šโˆ’1 . The transformation A๐‘š transforms the intermediate result of M๐‘šโˆ’1 โ—ฆ Aโˆ’1 ๐‘š , which is in the (๐‘ž ๐‘šโˆ’1 , ๐‘ ๐‘šโˆ’1 ) phase space, back to the new phase space in (๐‘ž ๐‘š , ๐‘ ๐‘š ). The nonlinear normal form transformation steps below are calculated up to third order. It will become obvious during the process that transformations of higher even and odd orders follow the same pattern as the second and third order transformation, respectively. 3.3.1 The Parameter Dependent Fixed Point The DA normal form algorithm starts with an origin preserving map. Accordingly, the result from the integration is shifted to the equilibrium/fixed point MFP = M0 โˆ’ C , hence MFP = L + ๐‘š U๐‘š ร is an origin preserving fixed point map with MFP ( 0) ยฎ = 0.ยฎ If the map were dependent on changes ๐›ฟ๐œ‚ of a system parameter ๐œ‚, e.g., changes in the driving frequency ๐œ” = ๐œ”0 + ๐›ฟ๐œ”, the normal form algorithm would require the calculation of the parameter dependent fixed point ๐‘งยฎ(๐›ฟ๐œ‚) = (๐‘ž FP (๐›ฟ๐œ‚), ๐‘ FP (๐›ฟ๐œ‚)) such that MFP ( 0, ยฎ ๐›ฟ๐œ‚) = 0. ยฎ In Eq. (3.3), the relation of the equilibrium point (fixed point) and the driving frequency was already calculated yielding the parameter dependent fixed point ! ! 1 ๐‘งยฎ (๐›ฟ๐œ”) = arccos ,0 for (๐œ”0 + ๐›ฟ๐œ”) 2 โ‰ฅ 1. (๐œ”0 + ๐›ฟ๐œ”) 2 40 For less straightforward systems, one uses the following inversion method on the extended map ( MFP โˆ’ I๐‘งยฎ, I๐›ฟ๐œ‚ยฎ) to find the parameter dependent fixed point ๐‘งยฎ(๐›ฟ๐œ‚) ยฎ [19, Eq. (7.47)]:     โˆ’1   ยฎ , I๐›ฟ๐œ‚ยฎ = MFP โˆ’ I๐‘งยฎ, I๐›ฟ๐œ‚ยฎ ๐‘งยฎ (๐›ฟ๐œ‚) ยฎ 0, ๐›ฟ๐œ‚ยฎ , (3.13) where I๐‘งยฎ and I๐›ฟ๐œ‚ยฎ are the identity map of ๐‘งยฎ and ๐›ฟ๐œ‚, ยฎ respectively. Given the parameter dependent fixed point, the map is expanded around it: MPDFP = MFP (ยฎ๐‘ง (๐›ฟ๐œ‚) ยฎ โˆ’ MFP (ยฎ๐‘ง (๐›ฟ๐œ‚) ยฎ + ๐‘งยฎ, ๐›ฟ๐œ‚) ยฎ , ๐›ฟ๐œ‚) ยฎ . (3.14) To limit the complexity of the walk-through of the DA normal form algorithm, we will not consider parameter dependence in the further calculations of this chapter and therefore proceed with MFP . 3.3.2 The Linear Transformation The first order transformation is the diagonalization, transforming the system into the eigenvector space of the linear part L. In order to determine the transformation A1 and its inverse Aโˆ’1 1 for the diagonalization, we determine the eigenvalues ๐œ† ยฑ and eigenvectors ๐‘ฃยฎยฑ of the linear matrix ๐ฟห† in the linear part L. For this, we require that all eigenvalues of MFP are distinct. Furthermore, we only consider cases where MFP is linearly stable, which means that all eigenvalues have an absolute value |๐œ†| โ‰ค 1. This also means that det( ๐ฟ) ห† โ‰ค 1, otherwise at least one of the eigenvalues is larger than 1, making the system linearly unstable. Particularly interesting is the case det( ๐ฟ) ห† = 1, which indicates that the system is symplectic and only stable in the case of complex conjugate eigenvalues ๐œ†ยฑ = ๐‘’ ยฑ๐‘–๐œ‡ . While there are procedures for the cases of real and degenerate eigenvalues with a magnitude smaller than one (see [19]), this chapter only illustrates the procedures for the most relevant and common symplectic case of only complex conjugate eigenvalues and eigenvectors. Solving the characteristic polynomial yields the eigenvalues โˆš๏ธ„ 2 ห† tr ๐ฟห†  tr ๐ฟ โˆ’ det ๐ฟห† = ๐‘Ÿ๐‘’ ยฑ๐‘–๐œ‡  ๐œ†ยฑ = ยฑ 2 4 ! โˆš๏ธƒ tr ๐ฟห† with ๐‘Ÿ = det ๐ฟห†  and ๐œ‡ = sign (๐‘„| ๐‘ 0 ) arccos . 2๐‘Ÿ 41 To generalize the procedure of diagonalization, the Twiss parameters [32] are used with (๐‘„|๐‘ž 0 ) โˆ’ (๐‘ƒ| ๐‘ 0 ) (๐‘„| ๐‘ 0 ) โˆ’(๐‘ƒ|๐‘ž 0 ) ๐›ผ= , ๐›ฝ= , and ๐›พ= . 2๐‘Ÿ sin ๐œ‡ ๐‘Ÿ sin ๐œ‡ ๐‘Ÿ sin ๐œ‡ With this notation the linear matrix ๐ฟห† can be generally written as ยฉcos ๐œ‡ + ๐›ผ sin ๐œ‡ ๐›ฝ sin ๐œ‡ ยช ๐ฟห† = ๐‘Ÿ ยท ยญยญ ยฎ. ยฎ โˆ’๐›พ sin ๐œ‡ cos ๐œ‡ โˆ’ ๐›ผ sin ๐œ‡ ยซ ยฌ The complex conjugate eigenvectors ๐‘ฃยฎยฑ associated with the complex conjugate eigenvalues ๐œ† ยฑ of ๐ฟห† are then obtained by solving ๐ฟห† โˆ’ ๐œ† ยฑ I ๐‘ฃยฎยฑ = 0. ยฎ  As a result, the following eigenvectors are calculated ยฉ ๐›ฝ ยช ยฉ๐›ผ ยฑ ๐‘– ยช ๐‘ฃยฎยฑ = ยญยญ ยฎ ยฎ or ๐‘ฃยฎยฑ = ยญยญ ยฎ, ยฎ โˆ’๐›ผ ยฑ ๐‘– โˆ’๐›พ ยซ ยฌ ยซ ยฌ for the case that either ๐›ฝ = 0 or ๐›พ = 0. The transformation Aโˆ’1 1 consist of the two complex conjugate eigenvectors ๐‘ฃยฎยฑ , guaranteeing that Aโˆ’1 1 (๐‘ž 1 , ๐‘ 1 ) is real just like the original variables (๐‘ž 0 , ๐‘ 0 ) and the fixed point map MFP . The transformation A1 is calculated accordingly such that the resulting map M1 = A1 โ—ฆ MFP โ—ฆ Aโˆ’1 1 is in the complex conjugate eigenvector space and has complex conjugate components Mฬ„+1 = Mโˆ’ 1 . For ๐›ฝ โ‰  0, the transformations are ยฉ (๐‘ž 0 |๐‘ž 1 ) (๐‘ž 0 | ๐‘ 1 ) ยช 1 ยฉยญ ๐›ฝ ๐›ฝ ยช Aโˆ’1 1 = ยญ ยญ ยฎ= โˆš ยฎ 2 ๐›ฝยญ ยฎ ยฎ and (3.15) ( ๐‘ 0 |๐‘ž 1 ) ( ๐‘ 0 | ๐‘ 1 ) ๐‘– โˆ’ ๐›ผ โˆ’๐‘– โˆ’ ๐›ผ ยซ ยฌ ยซ ยฌ ยฉ (๐‘ž 1 |๐‘ž 0 ) (๐‘ž 1 | ๐‘ 0 ) ยช ยฉโˆ’๐‘– โˆ’ ๐›ผ โˆ’๐›ฝยช A1 = ยญยญ ยฎ = โˆš๐‘– ยญ ยฎ. (3.16) ยฎ ๐›ฝ ยญ ยฎ ( ๐‘ |๐‘ž ) ( ๐‘ 1 | ๐‘ 0 ) โˆ’๐‘– + ๐›ผ ๐›ฝ ยซ 1 0 ยฌ ยซ ยฌ โˆš For the centrifugal governor example with ๐œ” = 2, the eigenvalues are ๐œ† ยฑ = ๐‘Ÿ๐‘’ ยฑ๐‘–๐œ‡ with ๐‘Ÿ = 1 and ๐œ‡ = 1.22474487. (3.17) The Twiss parameters are โˆš๏ธ‚ โˆš๏ธ‚ 2 3 ๐›ผ = 0, ๐›ฝ = 0.816496581 โ‰ˆ , and ๐›พ = 1.22474487 โ‰ˆ . 3 2 42 The resulting diagonalized map is of the form M1 = R + ๐‘š ๐‘† ๐‘š , where S๐‘š are the transformed ร nonlinear parts of order ๐‘š in the eigenvector space of ๐ฟห† and R is the diagonalized linear part, where the linear matrix ๐‘…ห† of R only consist of the eigenvalues ๐‘’ ยฑ๐‘–๐œ‡ on its main diagonal: ยฉ๐‘’๐‘–๐œ‡ + 0 ยช ยฉ ๐‘ž 1 ยช ยฉS2(2,0) ยช 2 ยฉS2(1,1) + ยช + ยฉS2(0,2) ยช 2 M1 (๐‘ž 1 , ๐‘ 1 ) = ยญ ยญ ยฎ ยญ ยฎ + ยญ ยฎ ๐‘ž ยฎ 1 ยญ โˆ’ + ยญ ยฎ ๐‘ž ๐‘ ยฎ 1 1 ยญ โˆ’ + ยญ ยฎ๐‘ ยฎ 1 โˆ’๐‘–๐œ‡ ยฎยญ ยฎ ยญ โˆ’ 0 ๐‘’ ๐‘1 S2(2,0) S2(1,1) S2(0,2) ยซ ยฌยซ ยฌ ยซ ยฌ ยซ ยฌ ยซ ยฌ | {z } | {z } R S2 ยฉS + ยฉS + ยฉS + ยฉS + + ยญยญ 3(3,0) ยฎยฎ ๐‘ž 31 + ยญยญ 3(2,1) ยฎยฎ ๐‘ž 21 ๐‘ 1 + ยญยญ 3(1,2) ยฎยฎ ๐‘ž 1 ๐‘ 21 + ยญยญ 3(0,3) ยฎยฎ ๐‘ 30 +... ยช ยช ยช ยช (3.18) Sโˆ’ Sโˆ’ Sโˆ’ Sโˆ’ ยซ 3(3,0) ยฌ ยซ 3(2,1) ยฌ ยซ 3(1,2) ยฌ ยซ 3(0,3) ยฌ | {z } S3 Tab. 3.3 lists the values to the coefficients above for the centrifugal governor example for a โˆš rotation frequency corresponding to an equilibrium angle of ๐œ™0 (๐œ” = 2) = ๐œ‹/3 = 60ยฐ. Table 3.3: Coefficients of M1 up to order three. Note the complex conjugate property S๐‘š(๐‘˜ ยฑ = + ,๐‘˜ โˆ’ ) โˆ“ Sฬ„๐‘š(๐‘˜ ,๐‘˜ ) . โˆ’ + Order Coeff. Real Part Imaginary Part 1 e๐‘–๐œ‡ 0.339185989 0.940719334 1 eโˆ’๐‘–๐œ‡ 0.339185989 -0.940719334 2 + S2(2,0) -0.216977793 -0.059191831 2 โˆ’ S2(2,0) 0.072325931 -0.102961500 2 + S2(1,1) -0.258557455 0.368076331 2 โˆ’ S2(1,1) -0.258557455 -0.368076331 2 + S2(0,2) 0.072325931 0.102961500 2 โˆ’ S2(0,2) -0.216977793 0.059191831 3 + S3(3,0) 0.068036138 0.047162997 3 โˆ’ S3(3,0) -0.045160062 -0.016282923 3 + S3(2,1) 0.259415349 -0.130475661 3 โˆ’ S3(2,1) -0.022283986 0.239186527 3 + S3(1,2) -0.022283986 -0.239186527 3 โˆ’ S3(1,2) 0.259415349 0.130475661 3 + S3(0,3) -0.045160062 -0.016282923 3 โˆ’ S3(0,3) 0.068036138 -0.047162997 43 3.3.3 The Nonlinear Transformations The nonlinear transformations are the key steps of the normal form algorithm. In this first part of this subsection, we are going to look at an ๐‘šth order transformation in general, before going through the nonlinear transformation for orders two and three in detail. 3.3.3.1 General ๐‘šth Order Nonlinear Transformation All the following nonlinear transformation steps are done order by order and are all of the same form: M๐‘š = A๐‘š โ—ฆ M๐‘šโˆ’1 โ—ฆ Aโˆ’1 ๐‘š , where the ๐‘šth transformation does not change any of the lower order terms of M๐‘šโˆ’1 that have already been transformed in the previous transformations. Hence, M๐‘š differs from M๐‘šโˆ’1 only in the orders ๐‘š and larger. The ๐‘šth order transformation A๐‘š = I + T๐‘š + Oโ‰ฅ๐‘š+1 , specifically the polynomial T๐‘š of only ๐‘šth order terms, is chosen such that the ๐‘šth order terms S๐‘š of the map M๐‘šโˆ’1 are simplified or even eliminated. Effects on the higher orders of M๐‘š due to the ๐‘šth order transformation can only be considered by adjusting the terms of order higher than ๐‘š of A๐‘š , namely Oโ‰ฅ๐‘š+1 . In other words, finding T๐‘š is essential to the DA normal form algorithm, while the terms Oโ‰ฅ๐‘š+1 can be chosen freely, e.g., to make the transformation symplectic by choosing A๐‘š = exp(๐ฟ T๐‘š ) or to avoid higher order resonances. Usually, the symplectic transformation is chosen since the calculation of the transformation A๐‘š and its inverse are straightforward. The flow operator ๐ฟ T๐‘š = ( T๐‘š+ ๐œ•๐‘ž + T๐‘šโˆ’ ๐œ•๐‘ ) in the exponential behaves in the following way:     0 1 1 2 exp ๐ฟ T๐‘š I = ๐ฟ T + ๐ฟ T + ๐ฟ T + O>(๐‘š+1) I ๐‘š ๐‘š 2 ๐‘š   + โˆ’ 1 + โˆ’ = 1 + ( T๐‘š ๐œ•๐‘ž + T๐‘š ๐œ•๐‘ ) + ๐ฟ T๐‘š ( T๐‘š ๐œ•๐‘ž + T๐‘š ๐œ•๐‘ ) + O>(๐‘š+1) (๐‘ž, ๐‘)๐‘‡ 2 1 = I + T๐‘š + ๐ฟ T๐‘š T๐‘š + O>(๐‘š+1) . (3.19) 2 So, the inverse is given by   1 Aโˆ’1 ๐‘š = exp โˆ’๐ฟ T๐‘š = I โˆ’ T๐‘š + ๐ฟ T๐‘š T๐‘š โˆ’ O>(๐‘š+1) . (3.20) 2 44 In the example case of the centrifugal governor, we investigate the DA normal form algorithm up to order three, which means for ๐‘š = 3:   A3 = exp ๐ฟ T3 I =3 I + T3 (3.21)   Aโˆ’13 = exp โˆ’๐ฟ T3 I =3 I โˆ’ T3 . (3.22) For the second order transformation it is necessary to consider the third order terms O3 , since they influence the third order terms of M2 :   A2 = exp ๐ฟ T2 I =3 I + T2 + O3 (3.23)   Aโˆ’1 2 = exp โˆ’๐ฟ T I =3 I โˆ’ T2 + O3 , 2 (3.24) with 1 1 O3 = ๐ฟ T๐‘š T๐‘š = ( T2+ ๐œ•๐‘ž + T2โˆ’ ๐œ•๐‘ ) T2 . (3.25) 2 2 As introduced in Sec. 2.1, the notation โ€˜=๐‘š โ€™ indicates that the quantities on both sides are equal up to expansion order ๐‘š. In order to determine T๐‘š , we analyze the ๐‘šth order transformation and only look at terms up to order ๐‘š [19, Eq. (7.62)]: A๐‘š โ—ฆ M๐‘šโˆ’1 โ—ฆ Aโˆ’1 ๐‘š =๐‘š ( I + T๐‘š ) โ—ฆ ( R + S๐‘š ) โ—ฆ ( I โˆ’ T๐‘š ) =๐‘š ( I + T๐‘š ) โ—ฆ ( R โˆ’ R โ—ฆ T๐‘š + S๐‘š ) =๐‘š R + S๐‘š + [ T๐‘š , R] . (3.26) Various terms with orders higher than ๐‘š are ignored in the equations above. The goal is to choose T๐‘š such that the commutator [ T๐‘š , R] = T๐‘š โ—ฆ R โˆ’ R โ—ฆ T๐‘š = โˆ’S๐‘š to simplify M๐‘š , i.e. the result of Eq. (3.26). The polynomials in the upper and lower component of T๐‘š can be express as โˆ‘๏ธ T๐‘šยฑ (๐‘ž, ๐‘) = ยฑ T๐‘š(๐‘˜ ๐‘˜+ ๐‘˜โˆ’ ,๐‘˜ ) ๐‘ž ๐‘ . (3.27) + โˆ’ ๐‘š=๐‘˜ + +๐‘˜ โˆ’ ๐‘˜ ยฑ โˆˆN0 Hence, the commutator C๐‘š = [ T๐‘š , R] yields โˆ‘๏ธ   ยฑ C๐‘š (๐‘ž, ๐‘) = ยฑ T๐‘š(๐‘˜ ,๐‘˜ ) ๐‘’ ๐‘–๐œ‡(๐‘˜ + โˆ’๐‘˜ โˆ’ ) โˆ’๐‘’ ยฑ๐‘–๐œ‡ ๐‘ž ๐‘˜+ ๐‘ ๐‘˜โˆ’ . (3.28) + โˆ’ ๐‘š=๐‘˜ + +๐‘˜ โˆ’ ๐‘˜ ยฑ โˆˆN0 45 A term in S๐‘š can only be removed if and only if the corresponding term in the commutator C๐‘š is not zero. Terms of the commutator are zero, whenever the condition ๐‘’๐‘–๐œ‡(๐‘˜ + โˆ’๐‘˜ โˆ’ ) โˆ’ ๐‘’ ยฑ๐‘–๐œ‡ = 0 (3.29) is satisfied, which is the case for ๐‘˜ + โˆ’ ๐‘˜ โˆ’ = ยฑ1. This (Eq. (3.29)) is the key condition of the DA normal form algorithm, since it determines the surviving nonlinear terms S๐‘š . All other terms that do not satisfy the condition are eliminated by choosing the coefficients of T๐‘š as follows โˆ’S๐‘š(๐‘˜ ยฑ ยฑ + ,๐‘˜ โˆ’ ) T๐‘š(๐‘˜ = . (3.30) + ,๐‘˜ โˆ’ ) ๐‘’ ๐‘–๐œ‡(๐‘˜ + โˆ’ ) โˆ’ ๐‘’ ยฑ๐‘–๐œ‡ โˆ’๐‘˜ Specifically, this means that the terms S๐‘š(๐‘˜,๐‘˜โˆ’1) + โˆ’ and S๐‘š(๐‘˜โˆ’1,๐‘˜) always survive for all uneven orders ๐‘š with ๐‘š = ๐‘˜ + ๐‘˜ โˆ’ 1 = 2๐‘˜ โˆ’ 1. 3.3.3.2 Explicit Second Order Nonlinear Transformation The polynomial T๐‘š from Eq. (3.27) for ๐‘š = 2 yields       T2 (๐‘ž, ๐‘) = T2ยฑ |2, 0 ๐‘ž 2 + T2ยฑ |1, 1 ๐‘ž ๐‘ + T2ยฑ |0, 2 ๐‘ 2 ยฉT + ยช ยฉT + ยช ยฉT + ยช = ยญยญ 2(2,0) ยฎยฎ ๐‘ž 2 + ยญยญ 2(1,1) ยฎยฎ ๐‘ž ๐‘ + ยญยญ 2(0,2) ยฎยฎ ๐‘ 2 . (3.31) T โˆ’ T โˆ’ T โˆ’ ยซ 2(2,0) ยฌ ยซ 2(1,1) ยฌ ยซ 2(0,2) ยฌ The commutator C2 = [ T2 , R] = T2 โ—ฆ R โˆ’ R โ—ฆ T2 of the second order nonlinear transformation has only nonzero terms with C2 (๐‘ž, ๐‘) = [T2 , R] (๐‘ž, ๐‘) = ( T2 โ—ฆ R โˆ’ R โ—ฆ T2 ) (๐‘ž, ๐‘)       = T2 |2, 0 ๐‘’ ๐‘ž + T2 |1, 1 ๐‘ž ๐‘ + T2 |0, 2 ๐‘’ โˆ’2๐‘–๐œ‡ ๐‘ 2 โˆ’ ๐‘’ ยฑ๐‘–๐œ‡ T2ยฑ (๐‘ž, ๐‘) ยฑ 2๐‘–๐œ‡ 2 ยฑ ยฑ       ยฉT + 2๐‘–๐œ‡ ๐‘’ โˆ’๐‘’ ๐‘–๐œ‡ ยฉT + 1โˆ’๐‘’ ๐‘–๐œ‡ ยฉT + ๐‘’ โˆ’2๐‘–๐œ‡ โˆ’๐‘’ ๐‘–๐œ‡ = ยญยญ 2(2,0)   ยฎยฎ ๐‘ž 2 + ยญยญ 2(1,1)   ยฎยฎ ๐‘ž ๐‘ + ยญยญ 2(0,2)   ยฎยฎ ๐‘ 2 ยช ยช ยช Tโˆ’ ๐‘’ 2๐‘–๐œ‡ โˆ’ ๐‘’ โˆ’๐‘–๐œ‡ Tโˆ’ 1 โˆ’ ๐‘’ โˆ’๐‘–๐œ‡ Tโˆ’ ๐‘’ โˆ’2๐‘–๐œ‡ โˆ’ ๐‘’ โˆ’๐‘–๐œ‡ ยซ 2(2,0) ยฌ ยซ 2(1,1) ยฌ ยซ 2(0,2) ยฌ (3.32) eliminating all S2 terms by choosing ยฑ โˆ’๐‘†2(๐‘˜ ยฑ + ,๐‘˜ โˆ’ ) T2(๐‘˜ = , (3.33) + ,๐‘˜ โˆ’ ) ๐‘’ ๐‘–๐œ‡(๐‘˜ + โˆ’๐‘˜ โˆ’ ) โˆ’ ๐‘’ ยฑ๐‘–๐œ‡ 46 since the condition from Eq. (3.29) is not satisfied: ๐‘’๐‘–๐œ‡(๐‘˜ + โˆ’๐‘˜ โˆ’ ) โˆ’ ๐‘’ ยฑ๐‘–๐œ‡ โ‰  0 โˆ€๐‘˜ + , ๐‘˜ โˆ’ โˆˆ N0 with ๐‘˜ + + ๐‘˜ โˆ’ = 2. The values of the T2(๐‘˜ ยฑ for the centrifugal governor example are given in Tab. 3.4. The + ,๐‘˜ โˆ’ ) terms of O3 are calculated via Eq. (3.25) from T2 and are also given in Tab. 3.4 yielding all terms of the transformation A2 and its inverse Aโˆ’1 2 from Eq. (3.23) and Eq. (3.24). Table 3.4: The values of the T2(๐‘˜ ยฑ ยฑ and O3(๐‘˜ . Note that T2 and O3 and therefore A2 and + ,๐‘˜ โˆ’ ) + ,๐‘˜ โˆ’ ) + its inverse are real with A๐‘š(๐‘˜ ,๐‘˜ ) = A๐‘š(๐‘˜ ,๐‘˜ ) . โˆ’ + โˆ’ โˆ’ + Order Coeff. Value Coeff. Value 2 + T2(2,0) -0.195635573 โˆ’ T2(2,0) 0.065211858 2 + T2(1,1) 0.391271145 โˆ’ T2(1,1) 0.391271145 2 + T2(0,2) 0.065211858 โˆ’ T2(0,2) -0.195635573 3 + O3(3,0) 0.051031036 โˆ’ O3(3,0) 0 3 + O3(2,1) -0.034020691 โˆ’ O3(2,1) 0.051031036 3 + O3(1,2) 0.051031036 โˆ’ O3(1,2) -0.034020691 3 + O3(0,3) 0 โˆ’ O3(0,3) 0.051031036 To study how the second order transformation affects the third order terms S3 of the map M2 , the transformation is considered up to third order: M2 =3 A2 โ—ฆ M1 โ—ฆ Aโˆ’1 2 =3 ( I + T2 + O3 ) โ—ฆ ( R + S2 + S3 ) โ—ฆ ( I โˆ’ T2 + O3 )   =3 ( I + T2 + O3 ) โ—ฆ R โ—ฆ ( I โˆ’ T2 + O3 ) + S2 โ—ฆ ( I โˆ’ T2 + O3 ) + S3 โ—ฆ ( I โˆ’ T2 + O3 ) ! z }| { z }| { =3 ( I + T2 + O3 ) โ—ฆ R โˆ’ R โ—ฆ T2 + R โ—ฆ O3 + S2 + S2โ†’3 +  Oโ‰ฅ4 + S3 +   O  โ‰ฅ4 =3 R โˆ’ R โ—ฆ T2 + R โ—ฆ O3 + S2 + S2โ†’3 + S3 + T2 โ—ฆ ( R โˆ’ R โ—ฆ T2 + R โ—ฆ O3 + S2 + S 2โ†’3 + S3 ) +  O  โ‰ฅ4 z }| { =3 T2 โ—ฆ R + K2โ†’3 +  O  +R โˆ’ R โ—ฆ T + R โ—ฆ O + S + S โ‰ฅ4 2 3 2 2โ†’3 + S3 =3 R + S2 + [ T2 โ—ฆ R] + S3 + S2โ†’3 + K2โ†’3 + R โ—ฆ O3 . (3.34) | {z } | {z } =0 S3,new 47 All the crossed-out terms  Oโ‰ฅ4 represent terms that do not contribute to the result up to order  three, since they are at least of order four. As a result of the second order transformation, the third order terms have changed and are summarized by S3,new . They are composed of the third order terms from after the linear transformation S3 and three new terms: S2โ†’3 =3 S2 โ—ฆ ( I โˆ’ T2 ) โˆ’ S2 , K2โ†’3 =3 T2 โ—ฆ ( R โˆ’ R โ—ฆ T2 + S2 ) โˆ’ T2 โ—ฆ R and R โ—ฆ O3 . While the last one is self-explanatory, the first two are not intuitively understood. In Sec. 3.3.3.4 these terms are calculated more explicitly, however, we recommend this section only for the very intrigued reader and encourage everyone else to skip it to follow the steps in the normal form algorithm. The result of the second order transformation M2 = R + S3,new for the example case of the centrifugal governor is given in Tab. 3.5. Table 3.5: New coefficients of third order of M2 after the second order transformation. Note that the first order terms remain unchanged and that the second order terms are all eliminated by the second order transformation. Interestingly, the second order transformation caused some terms of the third order to disappear in this specific case, which is not a general property of the second order transformation. The emphasized terms are surviving the third order transformation as explained in the following subsection. Order Coeff. Real Part Imaginary Part 3 + S3,new(3,0) 0.061270641 0.073920008 3 โˆ’ S3,new(3,0) 0 0 3 + S3,new(2,1) 0.470359667 -0.169592994 3 โˆ’ S3,new(2,1) 0 0.288035295 3 + S3,new(1,2) 0 -0.288035295 3 โˆ’ S3,new(1,2) 0.470359667 0.169592994 3 + S3,new(0,3) 0 0 3 โˆ’ S3,new(0,3) 0.061270641 -0.073920008 48 3.3.3.3 Explicit Third Order Nonlinear Transformation The third order transformation follows the same scheme as above (see Eq. (3.26)) only that the commutator C3 = [ T3 โ—ฆ R] has terms that are zero   T + ๐‘’ 3๐‘–๐œ‡ โˆ’ ๐‘’ ๐‘–๐œ‡ 0 C3 = ยญยญ 3(3,0)   ยฎยฎ ๐‘ž 3 + ยญยญ  ยฎยฎ ๐‘ž 2 ๐‘ ยฉ ยช ยฉ ยช  Tโˆ’ ๐‘’ 3๐‘–๐œ‡ โˆ’ ๐‘’ โˆ’๐‘–๐œ‡ Tโˆ’ ๐‘’๐‘–๐œ‡ โˆ’ ๐‘’ โˆ’๐‘–๐œ‡ ยซ 3(3,0) ยฌ ยซ 3(2,1) ยฌ     ยฉT + ๐‘’ โˆ’๐‘–๐œ‡ โˆ’ ๐‘’ ยช 2 ยฉ T3(0,3) ๐‘’ ๐‘–๐œ‡ + โˆ’3๐‘–๐œ‡ โˆ’๐‘’ ๐‘–๐œ‡ + ยญยญ 3(1,2)  ยฎยฎ ๐‘ 3 , ยช ยฎ ๐‘ž๐‘ + ยญ  (3.35) ยญ โˆ’ ๐‘’ โˆ’3๐‘–๐œ‡ โˆ’ ๐‘’ โˆ’๐‘–๐œ‡ ยฎ 0 T ยซ ยฌ ยซ 3(0,3) ยฌ + with C3(2,1) โˆ’ = C3(1,2) = 0. This means that the terms S3,new(2,1) + โˆ’ and S3,new(1,2) cannot be eliminated. All the other terms are eliminated by choosing ยฑ โˆ’๐‘†3,new(๐‘˜ ยฑ + ,๐‘˜ โˆ’ ) T3(๐‘˜ = for ๐‘˜ + โˆ’ ๐‘˜ โˆ’ โ‰  ยฑ1. (3.36) + ,๐‘˜ โˆ’ ) โˆ’๐‘˜ ) + โˆ’ โˆ’ ๐‘’ ยฑ๐‘–๐œ‡ ๐‘–๐œ‡(๐‘˜  ๐‘’ ยฑ The values of T3(๐‘˜ for the centrifugal governor example are given in Tab. 3.6. + ,๐‘˜ โˆ’ ) After the third order transformation the resulting map is of the following form ยฉ๐‘’๐‘–๐œ‡ 0 ยช ยฉ ๐‘ž 3 ยช ยฉS3,new(2,1) + 0 ยฎ ๐‘ž3 ๐‘2 ยช 2 ยฉ ยช M3 = ยญยญ ยฎยญ ยฎ+ยญ ยฎ ๐‘ž ๐‘3 + ยญ 3 3 0 ๐‘’ โˆ’๐‘–๐œ‡ ๐‘ 3 โˆ’ ยฎ ยญ ยฎ ยญ ยฎ ยญ ยฎ 0 S3,new(1,2) ยซ ยฌยซ ยฌ ยซ ยฌ ยซ ยฌ | {z } | {z } R S3,transformed   + ยฉ ๐‘’๐‘–๐œ‡ + S3,new(2,1) ๐‘ž 3 ๐‘ 3 ๐‘ž 3 ยช ยฉ ๐‘“ + (๐‘ž 3 ๐‘ 3 ) ๐‘ž 3 ยช = ยญ ยญ  ยฎยฎ = ยญยญ ยฎ. (3.37) ๐‘’ โˆ’๐‘–๐œ‡ + S3,new(1,2) โˆ’ ๐‘“ โˆ’ (๐‘ž 3 ๐‘ 3 ) ๐‘ 3 ยฎ ๐‘ž3 ๐‘3 ๐‘3 ยซ ยฌ ยซ ยฌ Table 3.6: The values of the T3(๐‘˜ ยฑ . The values for T3(2,1) + โˆ’ and T3(1,2) cannot be calculated + ,๐‘˜ โˆ’ ) because the denominator in Eq. (3.36) is zero. Note that T3(๐‘˜ ,๐‘˜ ) = T3(๐‘˜ ,๐‘˜ ) . + โˆ’ + โˆ’ โˆ’ + Order Coeff. Value Coeff. Value 3 + T3(3,0) 0.051031036 โˆ’ T3(3,0) 0 3 + T3(2,1) โ€“ โˆ’ T3(2,1) -0.153093109 3 + T3(1,2) -0.153093109 โˆ’ T3(1,2) โ€“ 3 + T3(0,3) 0 โˆ’ T3(0,3) 0.051031036 49 The corresponding values for the coefficients can be found in Tab. 3.3 for the linear terms and โˆ’ in Tab. 3.5 for the third order terms. The complex conjugate property of the map M+3 = M3 is maintained. While all nonlinear transformations follow the same structure, there is a fundamental difference between even and odd order transformation steps. For even order transformations there are no regularly surviving terms as shown for the second order transformation. For uneven order transformations, there are some terms of a special structure that do survive as shown for third order transformation. Higher even and odd order transformations will behave in the same way, and we will stop the process of the detailed walk-through here, after the third order transformation. In principle, the calculation of the transformations can be continued up to arbitrary order. With each transformation, the higher order terms are changed and in the end only the terms S๐‘š(๐‘˜,๐‘˜โˆ’1) + and โˆ’ S๐‘š(๐‘˜โˆ’1,๐‘˜) of uneven orders survive. Hence, the components Mยฑ ๐‘š can also be factorized into the ๐‘“ ยฑ (๐‘ž ๐‘š ๐‘ ๐‘š ) notation (see Eq. (3.37)) for higher orders. 3.3.3.4 The Effect of the Second Order Transformation on Third Order Terms The following calculation investigates the term S2โ†’3 as was previously done in [93] and was added here for sake of completeness. S2โ†’3 =3 S2 โ—ฆ ( I โˆ’ T2 ) โˆ’ S2  2  2    =3 S2(2,0) ๐‘ž โˆ’ T2+ + S2(0,2) ๐‘ โˆ’ T2โˆ’ + S2(1,1) ๐‘ž โˆ’ T2+ ๐‘ โˆ’ T2โˆ’ โˆ’ S2 ((( ( 2 ((((2 =3 S2(2,0) ๐‘ž(+ (S ((( (2(1,1) ๐‘ž ๐‘ + S2(0,2) ๐‘ โˆ’ S2 ( ( | ((( ( {z } =0 (  2 (((( ( ( 2 + S2(2,0) T2+ (( +( โˆ’+S โˆ’ ( ( +( S2(1,1) (((T( T 2 2 2(0,2) T 2 (( | ((( ( {z } โ‰ฅ O4   โˆ’ 2S2(2,0) T2+ ๐‘ž โˆ’ S2(1,1) T2+ ๐‘ + T2โˆ’ ๐‘ž โˆ’ 2S2(0,2) T2โˆ’ ๐‘. (3.38) As derived in the beginning of Sec. 3.3.3.3, the surviving parts of S2โ†’3 after the third order 50 + transformation are S2โ†’3(2,1) and its complex conjugate counterpart S2โ†’3(1,2) โˆ’ :   + S2โ†’3(2,1) = + โˆ’2S2(2,0) + T2(1,1) + โˆ’ S2(1,1) + T2(2,0) โˆ’ + T2(1,1) + โˆ’ 2S2(0,2) โˆ’ T2(2,0) + 2S2(2,0) + S2(1,1) + S2(1,1) + S2(2,0) + S2(1,1) โˆ’ S2(1,1) + S2(0,2) โˆ’ S2(2,0) = + + + . (3.39) 1 โˆ’ ๐‘’๐‘–๐œ‡ ๐‘’ 2๐‘–๐œ‡ โˆ’ ๐‘’๐‘–๐œ‡ 1 โˆ’ ๐‘’ โˆ’๐‘–๐œ‡ ๐‘’ 2๐‘–๐œ‡ โˆ’ ๐‘’ โˆ’๐‘–๐œ‡ This illustrates the complexity of these terms since every single term from S2 is relevant for them. Each term of S2 is again dependent on the terms of U2 . The relation is given by the linear transformation S2 = A1 โ—ฆ U2 โ—ฆ Aโˆ’1 1 . In principle, one can extend the calculation above to express + S2โ†’3(2,1) in terms of U2 and the Twiss parameters as done in [93]. The main insight however is that due to the significant influence of lower order transformation on higher order terms it is almost impossible to determine a priory which terms are the relevant ones for characteristics of the normal form. In the following calculation we are investigating the term K2โ†’3 , which was not previously investigated in [93]. K2โ†’3 = T2 โ—ฆ ( R โˆ’ R โ—ฆ T2 + S2 ) โˆ’ T2 โ—ฆ R = T2 โ—ฆ ( R โˆ’ K2 ) โˆ’ T2 โ—ฆ R  2  2 + =3 T2(2,0) ๐‘’ ๐‘ž โˆ’ K2 + T2(0,2) ๐‘’ ๐‘ โˆ’ K2 ๐‘–๐œ‡ โˆ’๐‘–๐œ‡ โˆ’    + โˆ’๐‘–๐œ‡ + T2(1,1) ๐‘’ ๐‘ž โˆ’ K2 ๐‘’ ๐‘ โˆ’ K2 โˆ’ T2 โ—ฆ R ๐‘–๐œ‡ โˆ’ (( ( ( (((( ๐‘ž ๐‘(+(T(2(0,2) ๐‘’ โˆ’2๐‘–๐œ‡ ๐‘ 2 โˆ’ T2 โ—ฆ R ( =3 T2(2,0) ๐‘’ 2๐‘–๐œ‡ ๐‘ž 2 + T(2(1,1) (( ( (( | (((( {z } (((( =0 (((  2 ( ( (((  2 + T2(2,0) K2+ (( +K(โˆ’( โˆ’ +(T ( (( K ( 2(1,1) 2 2 + T 2(0,2) K 2 (( | ((( ( {z } โ‰ฅ O4   โˆ’ 2T2(2,0) K2+ ๐‘’๐‘–๐œ‡ ๐‘ž โˆ’ T2(1,1) K2+ ๐‘’ โˆ’๐‘–๐œ‡ ๐‘ + K2โˆ’ ๐‘’๐‘–๐œ‡ ๐‘ž โˆ’ 2T2(0,2) K2โˆ’ ๐‘’ โˆ’๐‘–๐œ‡ ๐‘, (3.40) where K2 = R โ—ฆ T2 โˆ’ S2 โ†’ K2ยฑ = ๐‘’ ยฑ๐‘–๐œ‡ T2ยฑ โˆ’ S2ยฑ . (3.41) 51 So,   K2โ†’3 =3 2T2(2,0) S2+ ๐‘’๐‘–๐œ‡ ๐‘ž + T2(1,1) S2+ ๐‘’ โˆ’๐‘–๐œ‡ ๐‘ + S2โˆ’ ๐‘’๐‘–๐œ‡ ๐‘ž + 2T2(0,2) S2โˆ’ ๐‘’ โˆ’๐‘–๐œ‡ ๐‘   โˆ’ 2T2(2,0) T2 ๐‘’ ๐‘ž โˆ’ T2(1,1) T2 ๐‘ + T2 ๐‘ž โˆ’ 2T2(0,2) T2โˆ’ ๐‘’ โˆ’2๐‘–๐œ‡ ๐‘. + 2๐‘–๐œ‡ + โˆ’ (3.42) The surviving terms of K2โ†’3 after the third order transformation are K2โ†’3(2,1) + and its complex โˆ’ conjugate counterpart K2โ†’3(1,2) are   + K2โ†’3(2,1) = + 2T2(2,0) + S2(1,1) ๐‘’๐‘–๐œ‡ + + T2(1,1) + S2(2,0) ๐‘’ โˆ’๐‘–๐œ‡ โˆ’ + S2(1,1) ๐‘’๐‘–๐œ‡ + + 2T2(0,2) โˆ’ S2(2,0) ๐‘’ โˆ’๐‘–๐œ‡ โˆ’ 2T2(2,0) + + T2(1,1) ๐‘’ 2๐‘–๐œ‡   + + โˆ’ โˆ’ T2(1,1) T2(2,0) + T2(1,1) โˆ’ 2T2(0,2) + โˆ’ T2(2,0) ๐‘’ โˆ’2๐‘–๐œ‡ + โˆ’2S2(2,0) + S2(1,1) + S2(1,1)   + 2S2(0,2) โˆ’ S2(2,0) = โˆ’ S+ ๐‘’ โˆ’๐‘–๐œ‡ โˆ’ + S2(1,1) ๐‘’๐‘–๐œ‡ + ๐‘’๐‘–๐œ‡ โˆ’ 1 1 โˆ’ ๐‘’๐‘–๐œ‡ 2(2,0)  ๐‘’ 2๐‘–๐œ‡ โˆ’ ๐‘’ โˆ’๐‘–๐œ‡ + S2(1,1) + 2S2(2,0) โˆ’ + S2(1,1) + S2(1,1) + S2(2,0) + 2S2(0,2) โˆ’ S2(2,0) + โˆ’ โˆ’ . (3.43) 2 (cos ๐œ‡ โˆ’ 1) 2๐‘’ 2๐‘–๐œ‡ โˆ’ ๐‘’๐‘–๐œ‡ โˆ’ ๐‘’ 3๐‘–๐œ‡ 2๐‘’ 2๐‘–๐œ‡ โˆ’ ๐‘’ โˆ’๐‘–๐œ‡ โˆ’ ๐‘’ 5๐‘–๐œ‡ + Also for K2โ†’3(2,1) โˆ’ and K2โ†’3(1,2) the intertwine dependency on all terms of S2 becomes apparent highlighting the complex relation between lower order and higher order terms. 3.3.4 Transformation back to Real Space Normal Form Since the original map M0 only operates in real space, the normal form map MNF should also only operate in real space. This is why the current map M๐‘š , where ๐‘š is the order of last transformation, is transformed to a real normal form basis (๐‘ž NF , ๐‘ NF ) composed of the real and imaginary parts of the current complex conjugate basis (๐‘ž ๐‘š , ๐‘ ๐‘š ). Based on [19, Eq. (7.58) and (7.59) and (7.67)] the bases are related as follows ๐‘ž๐‘š + ๐‘๐‘š ๐‘ž๐‘š โˆ’ ๐‘๐‘š ๐‘ž NF = and ๐‘ NF = , (3.44) 2 2๐‘– and ๐‘ž ๐‘š = ๐‘ž NF + ๐‘– ๐‘ NF and ๐‘ ๐‘š = ๐‘ž NF โˆ’ ๐‘– ๐‘ NF with (3.45) ๐‘ž ๐‘š ๐‘ ๐‘š = ๐‘ž 2NF + ๐‘ 2NF = ๐‘Ÿ NF2 . (3.46) 52 The associated transfer matrix 1 ยฉยญ 1 1ยชยฎ ยฉยญ (๐‘ž NF |๐‘ž ๐‘š ) (๐‘ž NF | ๐‘ ๐‘š ) ยชยฎ Areal = = (3.47) 2 ยญโˆ’๐‘– ๐‘– ยฎ ยญ ( ๐‘ |๐‘ž ) ( ๐‘ | ๐‘ ) ยฎ ยซ ยฌ ยซ NF ๐‘š NF ๐‘š ยฌ to the real normal form basis is obtained from the equations above. The inverse relation is given by ยฉ1 ๐‘– ยช ยฉ (๐‘ž ๐‘š |๐‘ž NF ) (๐‘ž ๐‘š | ๐‘ NF ) ยช Aโˆ’1 = real ยญ ยญ ยฎ=ยญ ยฎ ยญ ยฎ. ยฎ (3.48) 1 โˆ’๐‘– ( ๐‘ ๐‘š |๐‘ž NF ) ( ๐‘ ๐‘š | ๐‘ NF ) ยซ ยฌ ยซ ยฌ The transformation back to the real space (into normal form space) yields   1 ยฉยญ 1 1ยชยฎ ยญ ยฉ ๐‘“+ 2 ๐‘Ÿ NF (๐‘ž NF + ๐‘– ๐‘ NF ) ยช MNF = Areal โ—ฆ M๐‘š โ—ฆ Aโˆ’1 real = ยทยญ   ยฎ 2 โˆ’๐‘– ๐‘– ยญ ยฎ โˆ’ 2 ๐‘“ ๐‘Ÿ NF (๐‘ž NF โˆ’ ๐‘– ๐‘ NF ) ยฎ ยซ ยฌ ยซ ยฌ ยฉ 21 ๐‘“ + + ๐‘“ยฏ+ ๐‘ž NF + 2๐‘– ๐‘“ + โˆ’ ๐‘“ยฏ+ ๐‘ NF ยช   = ยญยญ ยฎ โˆ’๐‘– ๐‘“ + โˆ’ ๐‘“ยฏ+  ๐‘ž 1 + ยฏ+  ยฎ 2 NF + 2 ๐‘“ + ๐‘“ ๐‘ NF ยซ ยฌ       ยฉRe ๐‘“ + ๐‘Ÿ NF 2 โˆ’Im ๐‘“ + ๐‘Ÿ NF 2 ยช ยฉ ๐‘ž NF ยช =ยญ   ยญ     ยฎยฎ ยท ยญยญ ยฎ. (3.49) + 2 + 2 ยฎ Im ๐‘“ ๐‘Ÿ NF Re ๐‘“ ๐‘Ÿ NF ๐‘ NF ยซ ยฌ ยซ ยฌ For the example of the centrifugal governor up to order three the normal form is     cos ๐œ‡ + 1 Re S + ๐‘Ÿ 2 โˆ’ sin ๐œ‡ โˆ’ 1 Im S + ๐‘Ÿ2 ยช ยฉ 2 3,new(2,1) NF 2 3,new(2,1) NF ยฎ ยญ ๐‘ž NF ยฎ ยฉ ยช MNF = ยญ ยญ     ยทยญ ยฎ. (3.50) 1 + 2 1 sin ๐œ‡ + 2 Im S3,new(2,1) ๐‘Ÿ NF cos ๐œ‡ + 2 Re S3,new(2,1) ๐‘Ÿ NF + 2 ยฎ ๐‘ ยซ ยฌ ยซ NF ยฌ The Tab. 3.7 below yields the values for the normal form map of our example case. Table 3.7: The normal form map MNF up to order three. The component M+NF is on the left, Mโˆ’ NF on the right. Order Coeff. Value Coeff. Value 1 M+NF(1,0) 0.339185989 Mโˆ’ NF(1,0) 0.940719334 1 M+NF(0,1) -0.940719334 Mโˆ’ NF(0,1) 0.339185989 3 M+NF(3,0) 0.470359667 Mโˆ’ NF(3,0) -0.169592994 3 M+NF(2,1) 0.169592994 Mโˆ’ NF(2,1) 0.470359667 3 M+NF(1,2) 0.470359667 Mโˆ’ NF(1,2) -0.169592994 3 M+NF(0,3) 0.169592994 Mโˆ’ NF(0,3) 0.470359667 53 The normal form transformation from M0 to MNF can be obtained by the combination of all the single transformations yielding MNF = Areal โ—ฆ A๐‘š โ—ฆ A๐‘šโˆ’1 โ—ฆ ... โ—ฆ A1 โ—ฆ AFP โ—ฆM0 | {z } A โ—ฆ Aโˆ’1 โˆ’1 โˆ’1 โˆ’1 FP โ—ฆ A1 โ—ฆ ... โ—ฆ A๐‘šโˆ’1 โ—ฆ A๐‘š โ—ฆ Areal . โˆ’1 (3.51) | {z } Aโˆ’1 The values of the coefficients of the full normal form transformation A are given in Tab. 3.8. Table 3.8: The normal form transformation A up to order three. The component A+ is on the left, Aโˆ’ on the right. Order Coeff. Value Coeff. Value 1 A+1(1,0) 1.106681920 Aโˆ’1(1,0) 0 1 A+1(0,1) 0 Aโˆ’1(0,1) -0.903602004 2 A+2(2,0) 0.319471552 Aโˆ’2(2,0) 0 2 A+2(1,1) 0 Aโˆ’2(1,1) 0.521694860 2 A+2(0,2) 0.425962069 Aโˆ’2(0,2) 0 3 A+3(3,0) -0.046111747 Aโˆ’3(3,0) 0 3 A+3(2,1) 0 Aโˆ’3(2,1) -0.414150918 3 A+3(1,2) -0.399635138 Aโˆ’3(1,2) 0 3 A+3(0,3) 0 Aโˆ’3(0,3) 0.025100056 Writing the complex conjugate functions ๐‘“ ยฑ from the equations above (particularly Eq. (3.49)) in a complex notation as   ยฑ๐‘–ฮ› ๐‘Ÿ 2   ๐‘“ยฑ 2 ๐‘Ÿ NF = ๐‘’ NF (3.52) illustrates circular behavior of the normal form:       ยฉcos ฮ› ๐‘Ÿ NF 2 โˆ’ sin ฮ› ๐‘Ÿ NF 2 ยช ยฉ ๐‘ž NF ยช MNF = ยญยญ       ยฎยฎ ยท ยญยญ ยฎ. (3.53) 2 2 ยฎ sin ฮ› ๐‘Ÿ NF cos ฮ› ๐‘Ÿ NF ๐‘ NF ยซ ยฌ ยซ ยฌ It shows that the normal form MNF consists of circular curves in phase space with only amplitude   dependent angle advancements ฮ› ๐‘Ÿ NF . 2 54 3.3.5 Invariant Normal Form Radius The squared normal form radius ๐‘Ÿ NF 2 is related to the original coordinates (๐‘ž , ๐‘ ) by the normal 0 0 form transformation A, where   2 (๐‘ž , ๐‘ ) = ๐‘ž 2 (๐‘ž , ๐‘ ) + ๐‘ 2 (๐‘ž , ๐‘ ) ๐‘Ÿ NF 0 0 NF 0 0 NF 0 0   = A2+ + A2โˆ’ (๐‘ž 0 , ๐‘ 0 ) . (3.54) Explicitly calculating the squared normal form radius with the normal form transformation A up to order three from Tab. 3.8 yields 2 = 1.224745๐‘ž 2 + 0.816497๐‘ 2 + 0.707107๐‘ž 3 ๐‘Ÿ NF (3.55) 3 0 0 0 โˆš๏ธ‚ โˆš๏ธ‚ 3 2 2 2 1 โ‰ˆ ๐‘ž0 + ๐‘ 0 + โˆš ๐‘ž 30 (3.56) 2 3 2 โˆš   2 2 ๐œ‹   ๐œ‹  =3 โˆš ๐ธ + ๐‘ž0, ๐‘0 โˆ’ ๐ธ , 0 , (3.57) 3 3 3 where ๐‘ โˆ’๐œ”2 sin2 ๐œ™ ๐ธ (๐‘ž, ๐‘) = + + (1 โˆ’ cos ๐œ™) (3.58) 2 2 can be straightforwardly derived from Eq. (3.4) and Eq. (3.6). This direct relationship between the energy ๐ธ, as an invariant or constant of motion, and the squared normal form radius up to order three confirms that the normal form radius constitutes a constant of motion up to calculation order. The invariant of motion is a family of functions that remain constant for all phase space states (๐‘ž, ๐‘) along their phase space motion. In particular, if ๐ผ (๐‘ž, ๐‘) is an invariant of motion, then so is ๐ผ 2 (๐‘ž, ๐‘) or any other function ๐‘“ (๐ผ), which is defined by the resulting values of ๐ผ. Furthermore, ๐ผ (๐‘„(๐‘ž, ๐‘), ๐‘ƒ(๐‘ž, ๐‘)) is also an invariant if (๐‘„, ๐‘ƒ) belong to the same phase space curve as (๐‘ž, ๐‘). Transfer maps can yield such relations (๐‘„(๐‘ž, ๐‘), ๐‘ƒ(๐‘ž, ๐‘)), since they can represent how a phase space final state (๐‘„, ๐‘ƒ) depends on the phase space initial state (๐‘ž, ๐‘). Accordingly, the energy ๐ธ and the normal form radius ๐‘Ÿ NF 2 are both functions of the same family and related by the transformations explained in the paragraph above. Up to order three, this relation includes a shift by a constant and scaling, but the relation might reveal itself to be more complex than this with higher orders. 55 3.3.6 Angle Advancement, Tune and Tune Shifts   In the beam physics terminology, the angle advancements ฮ› ๐‘Ÿ NF 2 are scaled to the interval [0, 1] instead of [0, 2๐œ‹] and referred to as the tune and amplitude dependent tune shifts [19]. The angle advancement can be calculated from the normal form map via +   ยฉ MNF ๐‘ =0 ยช     2 2 NF ยฎ + 2 ฮ› ๐‘Ÿ NF = ๐‘ž NF , ๐‘ NF = 0 = arccos ยญ ยฎ = arccos Re ๐‘“ ๐‘Ÿ NF (3.59) ยญ . ยญ ๐‘ž NF ยฎ ยซ ยฌ For the centrifugal governor example up to order three, the angle advancement is given by    2 2  1  +  2 ฮ› ๐‘Ÿ NF = ๐‘ž NF = arccos cos ๐œ‡ + Re S3,new(2,1) ๐‘Ÿ NF 2   + Re S3,new(2,1) = ๐œ‡โˆ’ 2 . ๐‘Ÿ NF (3.60) 2 sin ๐œ‡ Note that ๐œ‡ is the eigenvalue phase of the original linear part (see Eq. (3.17)). The tune and tune shifts are calculated from Eq. (2.31), with   2 ฮ› ๐‘Ÿ NF   2 = ๐œˆ ๐‘Ÿ NF = 0.1949242 โˆ’ 0.07957747๐‘Ÿ NF 2 , (3.61) 2๐œ‹ where the constant part is the tune already known from the linear transformation with ๐œ‡ 1.22474487 ๐œˆ= = = 0.1949242. (3.62) 2๐œ‹ 2๐œ‹ With the expression of ๐‘Ÿ NF 2 in terms of the original coordinates (๐‘ž , ๐‘ ) from Eq. (3.55) the 0 0 tune and tune shifts are evaluated to ๐œˆ (๐‘ž 0 , ๐‘ 0 ) = 0.1949242 โˆ’ 0.0974621๐‘ž 20 โˆ’ 0.0649747๐‘ 20 โˆ’ 0.05626977๐‘ž 30 . (3.63) โˆš This yields a key insight into the centrifugal governor behavior for ๐œ” = 2. We already know โˆš โˆš that the centrifugal governor is rotating at 2/(2๐œ‹) โ‰ˆ 0.225 revolutions per ๐‘‡0 for ๐œ” = 2. The tune of about 0.195 tells us that the centrifugal governor arms oscillate at a frequency of about 0.195 + ๐‘ oscillations per ๐‘‡0 around their equilibrium position. The negative tune shifts additionally show that this frequency is decreasing for increasing amplitude of oscillation. 56 Since the map can only compare initial and final state of the oscillation after the integration time of 1 ๐‘‡0 we only know how much the oscillation cycle has advanced over this period, but not how many additional full oscillations ๐‘ have been completed in the meantime. By doing the same โˆš โˆš process as above for the centrifugal governor with ๐œ” = 2 for a Poincarรฉ map after time ๐‘ก = 2๐œ‹/ 2, i.e. one full centrifugal governor revolution, yields ๐œˆ (๐‘ž 0 , ๐‘ 0 ) = 0.8660254 โˆ’ 0.4330127๐‘ž 20 โˆ’ 0.2886751๐‘ 20 โˆ’ 0.25000000๐‘ž 30 , (3.64) โˆš which is exactly a factor of 2๐œ‹/ 2 larger than the tunes from Eq. (3.63). This means that ๐‘ must be zero and we did not miss any full oscillations during the integration up to ๐‘ก = 1. From Eq. (3.63) we can directly calculate the period of oscillation from normal form ๐‘‡NF , which is just 1/๐œˆ (๐‘ž 0 , ๐‘ 0 ). To compare the calculated normal form period ๐‘‡NF to the actual period of oscillation, we flip the horizontal and vertical axis from Fig. 3.4 and overlay the oscillatory plot with the calculated periods (see Fig. 3.6). The centrifugal governor arms are initiated with multiple angle offsets with ๐‘ ๐œ™ = 0 relative to their equilibrium angle at ๐œ™0 = 60ยฐ. If the normal form calculation of the period 8 7 6 time [๐‘‡0 ] 5 Arm Oscillation 4 ๐‘‡NF O10 ๐‘‡NF O3 3 2 1 0 0 10 20 30 40 50 60 70 80 90 angle ๐œ™ of governor arms [deg] Figure 3.6: Comparison between the calculated period with normal form methods ๐‘‡NF = 1/๐œˆ(๐‘ž 0 , ๐‘ 0 ) for calculation order ten (O10) and calculation order three (O3) to the actual โˆš period of oscillation given by the oscillatory behavior of the centrifugal governor arms for ๐œ” = 2 from Fig. 3.4. 57 is correct, the calculated period will agree with the time when the equilibrium governor arms reach their initial position amplitude after one actual period of oscillation. The higher the amplitude of oscillation, the more relevant are higher order effects. Accordingly, the accuracy drops with larger amplitudes. The order three calculation performs well between 35ยฐ(๐›ฟ๐œ™ = โˆ’25ยฐ) and 75ยฐ(๐›ฟ๐œ™ = +15ยฐ), while the order ten calculation can extend an accurate description over the range from 25ยฐ(๐›ฟ๐œ™ = โˆ’35ยฐ) to 85ยฐ(๐›ฟ๐œ™ = +25ยฐ). The normal form algorithm can also be performed with parameters, e.g., depending on changes to ๐œ”. In Tab. 3.9 result for the amplitude and parameter ๐›ฟ๐œ” dependent tunes shifts are listed. It shows that the ๐›ฟ๐œ” dependent tune shifts are positive, which means that an increase in ๐œ” increases the oscillation frequency of centrifugal governor arms. This is related to the deeper potential well. This knowledge about the dependency of the tunes on parameter shifts can help by the selection of a suitable ๐œ”, e.g., to avoid resonances between the governors revolution frequency and the oscillation frequency of the arms. While such a resonance is irrelevant in this simplified example it might be critical when the governor is part of a more complex system. Table 3.9: Tune and coefficients โˆš of amplitude and parameter ๐›ฟ๐œ” dependent tune shifts for centrifugal governor with ๐œ”0 = 2. Exponents Exponents Coefficient ๐‘ž 0 ๐‘ 0 ๐›ฟ๐œ” Coefficient ๐‘ž 0 ๐‘ 0 ๐›ฟ๐œ” 0.1949242003 0 0 0 -0.0562697698 3 0 0 0.3355884937 0 0 1 -0.0307638305 2 0 1 -0.0974621002 2 0 0 0.1741334861 1 1 1 -0.0649747334 0 2 0 0.1123973696 0 2 1 0.1591549431 1 0 1 -0.0435458248 1 0 2 -0.5753522001 0 0 2 -0.0142179396 0 1 2 0.0866936204 0 0 3 3.4 Visualization of the Different Order Normal Forms and Conclusion In this chapter, we considered the system of a centrifugal governor with a fixed rotation frequency โˆš of ๐œ” = 2 and analyzed it using the DA normal form algorithm. To visualize the effect of the different steps in the DA normal form algorithm, Fig. 3.7 shows 58 phase space tracking pictures for incomplete normal form maps. Given the tenth order Poincarรฉ map โˆš which describes the behavior of the centrifugal governor for ๐œ” = 2, these incomplete normal form maps stopped the normal form transformations at an order ๐‘› < 10 such that the resulting incomplete normal form map is only normalized up to order ๐‘›. There is no practical use for these incomplete normal form maps other than showing the progress of the normal form algorithm, since to make use of the normal form properties completion of the normal form transformation to the full order of the map is required. The phase space behavior in the full order normal form with its rotationally invariant property was previously shown in Fig. 3.5. The difference between a) and b) in Fig. 3.7 shows the effect of the linear transformation, which scales the variables to create circles close to the expansion point. The nonlinear distortions for larger amplitudes are still present. With the second and third order transformation, these distortions are removed in the normal form, however still not forming perfect circles for larger amplitudes. As a result of the DA normal form algorithm, we were able to produce invariants of motion up to calculation order. Specifically, we could show how the squared normal form radius is directly related to the energy ๐ธ up to calculation order (see Eq. (3.57)), which is a constant of motion for this system. The normal form algorithm also provided transformations from the original coordinates to the normal form coordinates, which were used to relate the phase space amplitudes to the normal form invariant. Finally, the normal form produced the period of oscillation of the centrifugal governor arms around their equilibrium angle depending on the amplitude of oscillation. The preformed calculation of order ten did not capture all the relevant high order effects at vary large amplitudes. However, yet higher order calculation would describe the period of oscillation for these amplitudes more accurately. 59 a) b) c) d) Figure 3.7: Phase space tracking of incomplete normal โˆš form maps of order ten of the centrifugal governor arms with a fixed rotation frequency of ๐œ” = 2. The original map (a), only linear normal form transformation (b), and only normal form transformations up to order two (c) and three (d), respectively. The normal form up to the full tenth order was illustrated in Fig. 3.5. 60 CHAPTER 4 BOUNDED MOTION PROBLEM This chapter contains large parts of my paper Bounded motion design in the Earth zonal problem using differential algebra based normal form methods published in Celestial Mechanics and Dynamical Astronomy, Vol. 132, 14 (2020) [95]. The paper was authored by Roberto Armellin, Martin Berz, and me. Given the detailed understanding of the differential algebra (DA) normal form algorithm from Sec. 2.3 and Chapter 3, we present its application in a new technique for the calculation of entire continuous sets of orbits, which remain in long term relative bounded motion under zonal gravitational perturbation. We will see that the application of the DA normal form algorithm in this particular case is only possible due to a well-chosen Poincarรฉ surface for the Poincarรฉ return map (Sec. 2.2), which captures the critical phase space behavior at the right space-time instance, which requires a combination of dimension-reducing phase space projections. 4.1 Introduction to Bounded Motion The term โ€˜bounded motionโ€™ is used in the field of astrodynamics to describe a special orbital flight pattern of two objects (usually man-made satellites), where the two objects remain in close proximity to each other over an extended period of time. Both objects are on orbits around a common central gravitational body like a planet, moon, asteroid, or star, and their relative distance is bounded. In practice, bounded motion finds application in cluster flight [31] and formation flying [5] missions, which can offer many advantages compared to single spacecraft missions. From the scientific standpoint, they enable measurements of unprecedented spatial and temporal correlation, but they also have economic advantages such as allowing for redundancies within the spacecraft group, a distribution of the payload, and the adaptability of the mission by exchanging modules of the group. Missions such as PRISMA [33], GRACE [66], and TerraSAR-X and TanDEM-X [34] demonstrated the practicability of formation flying and stimulated further research in the field. 61 Moving from an ideal unperturbed system with elliptical Kepler orbits to the realistic mission case by considering perturbations to the dynamics makes it not trivial to find bounded motion orbits. The dominating perturbation is often due to the oblateness of the central body and the associated zonal perturbation from the second zonal harmonic coefficient ๐ฝ2 of the gravitational potential. This zonal perturbation introduces a drift in the right ascension of the ascending node (RAAN) ฮ”ฮฉ, the argument of periapsis, and the mean anomaly. The drift in each of the quantities is oscillating at different frequencies, which drastically increases the complexity of the bounded motion problem. Additional non-zonal gravitational perturbations break the rotational symmetry of the system and the regular oscillations in each of the quantities mentioned above, which complicates the problem even more. To minimize the extent of formation-keeping maneuvers with control strategies during a mission, it is of great interest to the astrodynamical community to find โ€˜naturallyโ€™ bounded motion orbits for models considering as many perturbations as possible, which leave only the unmodeled perturbations to be corrected by control maneuvers. In this chapter and in [95], we present a method that allows for the design of long term relative bounded motion considering a zonal gravitational model using normal form methods. Since [95] contains an extensive literature review of previous approaches, only contributions directly linked to our technique for the zonal problem will be mentioned below. The pioneering work by Broucke [30] on families of two dimensional quasi-periodic invariant tori around stable periodic orbits of the Ruth-reduced axially symmetric system was used by Koon et al. [46] in combination with Poincarรฉ section techniques to study the ๐ฝ2 problem. While this method improved first order approaches, long term bounded motion was still not achieved by placing orbits on the center manifold. Xu et al. [100] pointed out that long term bounded motion in the zonally perturbed system could only be achieved when the RAAN drift ฮ”ฮฉ and nodal period ๐‘‡๐‘‘ are on average the same for each of the bounded modules (see Sec. 4.2.5). These constraints are weaker than the constraints originally derived by Martinusi and Gurfil [65]. In [9], a fully numerical technique based on stroboscopic maps was used to obtain entire families of quasi-periodic orbits producing bounded relative motion about a periodic one. This method was 62 then used to study both: bounded motion about asteroids [8] and in low Earth, medium Earth, and geostationary orbits [10]. Numerical approaches yield bounded relative orbits with arbitrary size over very long periods of time (or infinite time in theory). However, they require complex and time-consuming algorithms. In [42], a compromise between the analytic and numerical technique was presented based on the use of DA. DA techniques were used to expand to high order the mapping between two consecutive equatorial crossings (i.e., Poincarรฉ maps). This enabled the study of the motion of a spacecraft for many revolutions by the fast evaluation of Taylor polynomials. The problem of designing bounded motion orbits was then reduced to the solution of two nonlinear polynomial equations, namely constraining the mean nodal period ๐‘‡๐‘‘ and drift of the right ascension of the ascending node ฮ”ฮฉ. The derived method showed an accuracy comparable with that of fully numerical methods but with a reduced complexity due to the introduced polynomial approximations. The main drawback of this technique consisted of the calculation of the mean ๐‘‡๐‘‘ and ฮ”ฮฉ using numerical averaging over thousands of nodal crossings. This process resulted in the computationally intensive part of the algorithm and was also responsible for accuracy degradation in the case of very large separations. The advantage of our approach is that it overcomes this limitation when calculating bounded motion orbits under zonal perturbation by the introduction of DA based normal form (DANF) methods. In particular, the high-order DANF algorithm is used to transform the Poincarรฉ map into normal form space, in which the phase space behavior is circular and can be easily parameterized by action-angle coordinates (see Fig. 4.3). The action-angle representation of the normal form coordinates is then used to parameterize the original phase space coordinates of the Poincarรฉ return map. The original map is averaged over a full phase space revolution by a path integral along the angle parameterization, yielding the Taylor expansion of the averaged bounded motion quantities ๐‘‡๐‘‘ and ฮ”ฮฉ, for which the bounded motion conditions are straightforwardly imposed. Sets of highly accurate bounded orbits are obtained in the full zonal problem, extending over several thousand kilometers and valid for decades. This method avoids the numerical averaging introduced in [42]. The superiority in terms of elegance, computational time, and accuracy of the new algorithm will be 63 demonstrated using similar test cases to those presented in [42] and [10]. Before introducing our approach from [95], we start with some basics on the orbital motion under gravitational perturbation. Later we will show our results for the full zonal problem [95]. 4.2 Understanding Orbital Motion Under Gravitational Perturbation We consider the orbital motion around a single central body of mass, where the motion is only determined by the gravitational potential of the central body. Perturbations due to atmospheric drag, solar radiation pressure, or the gravitational field of other space bodies are ignored. We also ignore parabolic and hyperbolic orbits, which escape the gravitational potential due to their large enough kinetic energy. 4.2.1 The Perturbed Gravitational Potential Any gravitational potential ๐‘ˆ can be expressed in terms of spherical harmonics ๐‘Œ๐‘™,๐‘š and the corresponding coefficients ๐‘˜ ๐‘™,๐‘š : โˆž โˆ‘๏ธ ๐‘™  ๐‘™ ! ๐œ‡ โˆ‘๏ธ ๐‘…0 ๐‘ˆ (๐‘Ÿ, ๐œƒ, ๐œ™) = โˆ’ 1 + ๐‘˜ ๐‘™,๐‘š ๐‘Œ๐‘™,๐‘š (๐œƒ, ๐œ™) , (4.1) ๐‘Ÿ ๐‘Ÿ ๐‘™=1 ๐‘š=โˆ’๐‘™ where (๐‘Ÿ, ๐œƒ, ๐œ™) are spherical coordinates with the origin at the center of mass and where ๐œ‡ is the product of the gravitational constant and the mass of the central body. The coefficients of the ๐‘Œ๐‘™,๐‘š are often split into ๐‘˜ ๐‘™,๐‘š ยท ๐‘…0๐‘™ to make them independent of the size ๐‘…0 of the central body. The orientation of the coordinate system is usually chosen such that ๐‘งห† (๐œƒ = 0) aligns with the dominating symmetry axis of the central body. The plane perpendicular to ๐‘งห†, i.e. the ๐‘ฅ๐‘ฆ plane or ๐œƒ = ๐œ‹/2 plane, is referred to as the equatorial plane. The spherical harmonics can be grouped into three categories. Zonal terms (๐‘š = 0) are independent of the longitude ๐œ™ creating zones in the vertical/latitudinal direction. Sectional terms (๐‘š = ๐‘™) on the other hand are independent of the latitude ๐œƒ creating sections longitudinally. Tesseral terms (0 < ๐‘š < ๐‘™) are dependent on both ๐œ™ and ๐œƒ creating a chessboard pattern on the sphere. Each 64 of these terms is considered a gravitational perturbation to the spherically symmetric potential ๐œ‡ ๐‘ˆ0 = โˆ’ , (4.2) ๐‘Ÿ which only depends on the distance ๐‘Ÿ. The gravitational potentials of many rotating central bodies are dominated by their low order zonal terms, in particular, ๐‘Œ2,0 , since centrifugal effects of the rotation often cause a zonally dependent mass distribution with more mass at the equator and less mass at the poles compared to the sphere. Considering only the effects of zonal perturbations is also referred to as the zonal problem and is going to be the basis of our analysis. The axial symmetry conserves the angular momentum component along the symmetry axis and simplifies the potential significantly as the spherical harmonics ๐‘Œ๐‘™,๐‘š reduce to the ordinary Legendre polynomials ๐‘ƒ๐‘™ , with โˆž  ๐‘™ ! ๐œ‡ โˆ‘๏ธ ๐‘…0 ๐‘ˆ (๐‘Ÿ, ๐œƒ) = โˆ’ 1 + ๐ฝ๐‘™ ๐‘ƒ๐‘™ (cos ๐œƒ) . (4.3) ๐‘Ÿ ๐‘Ÿ ๐‘™=1 4.2.2 The Equations of Motion To calculate the behavior of an object in the perturbed gravitational field, we derive the equations of motion, which describe the dynamics as a set of mathematical functions. To be consistent with previous approaches and [95], we will use cylindrical coordinates. The starting point of the derivation is the Lagrangian 1 2 2 2 ยค2  ๐ฟ= ๐œŒยค + ๐‘งยค + ๐œŒ ๐œ™ โˆ’ ๐‘ˆ (๐œŒ, ๐‘ง, ๐œ™) (4.4) 2 of the system in cylindrical coordinates (๐œŒ, ๐‘ง, ๐œ™), where ๐œŒ is the distance in the equatorial plane โˆš๏ธ such that ๐‘Ÿ = ๐œŒ 2 + ๐‘ง2 yields the total distance between the orbiting object and the center of mass. The potential takes the following form in cylindrical coordinates โˆž โˆ‘๏ธ " ๐‘™  # ๐‘…0 ๐‘™  ๐‘ง ๐œ‡ โˆ‘๏ธ  ๐‘ˆ (๐œŒ, ๐‘ง, ๐œ™) = โˆ’ 1 + ๐‘ƒ๐‘™,๐‘š ๐ถ๐‘™,๐‘š cos (๐‘š๐œ™) + ๐‘†๐‘™,๐‘š sin (๐‘š๐œ™) , (4.5) ๐‘Ÿ ๐‘Ÿ ๐‘Ÿ ๐‘™=1 ๐‘š=0 where ๐‘ƒ๐‘™,๐‘š are the associated Legendre polynomials. 65 For the zonal problem (๐‘š = 0), the ๐‘ƒ๐‘™,๐‘š reduce to the ordinary Legendre polynomials ๐‘ƒ๐‘™ . The coefficients ๐ถ๐‘™,0 of the zonal problem are often denoted by ๐ฝ๐‘™ .  The canonical momenta ๐‘ฃ ๐œŒ , ๐‘ฃ ๐‘ง , ๐‘ฃ ๐œ™ to the position variables (๐œŒ, ๐‘ง, ๐œ™) are given by ๐œ•๐ฟ ๐œ•๐ฟ ๐œ•๐ฟ ๐‘ฃ๐œŒ = = ๐œŒยค ๐‘ฃ๐‘ง = = ๐‘งยค ๐‘ฃ๐œ™ = = ๐œŒ 2 ๐œ™ยค  H๐‘ง , (4.6) ๐œ• ๐œŒยค ๐œ• ๐‘งยค ๐œ•๐œ™ยค where H๐‘ง is the angular momentum component along the symmetry axis ๐‘งห† and the canonical momentum to the angle ๐œ™. From the Lagrange-Euler equations it follows that ๐œ•๐‘ˆ Hยค ๐‘ง = โˆ’ , (4.7) ๐œ•๐œ™ which is zero for the zonal problem due to the axial symmetry making H๐‘ง a constant of motion. Using the Legendre transformation, the Hamiltonian ! H 2 1 2 ๐‘ฃ + ๐‘ฃ 2๐‘ง + ๐‘ง ๐ป= + ๐‘ˆ (๐œŒ, ๐‘ง, ๐œ™) (4.8) 2 ๐œŒ ๐œŒ2 is obtained. Due to the time independence of the system (d๐‘ก ๐ป = 0), the Hamiltonian is equivalent to the energy ๐ธ, which is a constant of motion. The equations of motion are derived from the Hamiltonian via the Hamilton equations H๐‘ง ๐œŒยค = ๐‘ฃ ๐œŒ ๐‘งยค = ๐‘ฃ ๐‘ง ๐œ™ยค = (4.9) ๐œŒ2 H2๐‘ง ๐‘‘๐‘ˆ ๐‘‘๐‘ˆ ๐‘‘๐‘ˆ ๐‘ฃยค ๐œŒ = โˆ’ ๐‘ฃยค ๐‘ง = โˆ’ Hยค ๐‘ง = โˆ’ . (4.10) ๐œŒ3 ๐‘‘๐œŒ ๐‘‘๐‘ง ๐‘‘๐œ™ The time evolution X (๐‘ก) of the state X = (๐‘Ÿ, ๐‘ฃ ๐‘Ÿ , ๐‘ง, ๐‘ฃ ๐‘ง , ๐œ™, H๐‘ง )๐‘‡ of a spacecraft is determined by integrating the system of ODEs Xยค = ๐‘“ ( X ) from above. The orbit O of the spacecraft is described by the set of all states X (๐‘ก). 4.2.3 The Kepler Orbit Before we investigate the orbital behavior under perturbation, it is advisable to understand the unperturbed system with the spherically symmetric gravitational potential ๐‘ˆ0 . The orbiting motion of an object in the unperturbed potential takes the Keplerian form of a closed ellipse, which makes the motion two dimensional. The plane in which the ellipse lies is called the orbital plane. 66 The traditional orbital elements (๐‘Ž, ๐‘’, ๐‘–, ฮฉ, ๐œ”, ๐œˆ(๐‘ก)), also called Keplerian elements, characterize the position and orbit of the object using the elliptical shape as well as the equatorial plane of the central body as a reference (see [5] for a detailed description). The variables ๐‘Ž and ๐‘’ define the size (semi-major axis) and shape (eccentricity) of the ellipse, respectively. To describe the orientation of the orbital plane with respect to the central body, the reference direction ๐‘ฅห† within the equatorial plane is defined. Except for orbits within the equatorial plane, the elliptical orbit intersects with the equatorial plane in two places. The intersection in the ๐‘งห† direction (from south to north) is called the ascending node . The angle between the equatorial plane and the orbital plane is called the inclination ๐‘–. The angle between the reference direction ๐‘ฅห† and the ascending node within the equatorial plane is the longitude or right ascension of the ascending node (RAAN) ฮฉ. The argument of periapsis ๐œ” describes the orientation of the ellipse within the orbital plane as the angle between the ascending node and the periapsis (closest point of the ellipse to the origin). The true anomaly ๐œˆ(๐‘ก) yields the position of the object along the ellipse as the angle between the periapsis and the object. The time between two consecutive ascending nodes is called the nodal   period ๐‘‡๐‘‘ , with ๐‘‡๐‘‘ = ๐‘ก ( ๐‘›+1 ) โˆ’ ๐‘ก ( ๐‘› ). 4.2.4 Orbits Under Gravitational Perturbation The elliptical orbits deform under gravitational perturbations such that the orbits no longer close after a revolution around the central body. The description of perturbed orbits using Keplerian elements has to be carefully considered, since the four elements (๐‘Ž, ๐‘’, ๐œ”, ๐œˆ) are based on the assumption of an elliptical orbit in an unperturbed system. The elements ๐‘– and ฮฉ, on the other hand, only describe the orientation of the orbital plane, determined by position and velocity vectors of the orbiting object, but make no assumptions about the shape of the orbit. In practice, the Keplerian elements are calculated at each point in time assuming the orbit is an ellipse in an unperturbed system while propagating the object in the perturbed system. This representation is particularly helpful when the gravitational potential is only slightly 67 perturbed. It shows how the unperturbed elliptical orbit is influenced by the perturbations at each point in time. In Fig. 4.1, the orbital elements of a low Earth orbit (O2 from Sec. 4.4.1) under zonal perturbation are shown. As a reference, the orbit is also initiated with the same starting conditions but propagated considering only the spherically symmetrical part of the Earthโ€™s gravitational field. 1.141 ๐‘Ž [๐‘…0 ] 1.14 1.139 1.138 1.137 0.116 ๐‘’ 0.114 0.112 1.724 ๐‘– [rad] 1.7239 1.7238 perturbed 2๐œ‹ ฮฉ [rad] 0.002 unperturbed 0.001 ๐œ‹ 0 0 3.14 2๐œ‹ ๐œ” [rad] 3.13 ๐œ‹ 3.12 3.11 0 0 50 100 150 200 0 50 100 150 200 time [min] time [days] Figure 4.1: The behavior of the Keplerian elements of a low Earth orbit under zonal gravitational perturbations up to ๐ฝ15 (purple) and as a regular Kepler orbit in the unperturbed gravitational field (green) over time. Left and right plots show different time scales of the behavior. Note that the vertical scales of ฮฉ and ๐œ” are adjusted in the right plot to show the long term behavior. Compared to the unperturbed motion, the behavior of the Keplerian elements under zonal perturbation is quite complex. There are multiple oscillations happening at different frequencies. On the short time scale (see left plots in Fig. 4.1), there is the semi-periodic behavior associated with one orbital revolution with a nodal period of roughly 103 min. As already mentioned in the introduction, the zonal perturbation introduces a drift of the orbital plane, which is indicated by the increasing ฮฉ in Fig. 4.1. The corresponding long term behavior suggests that the orbital plane is rotating around the symmetry axis in about 365 days. However, as we will discover in Sec. 4.4.1 and in particular in Fig. 4.4 neither the nodal period ๐‘‡๐‘‘ nor the drift in the ascending node are constant, 68 but they are also oscillating. The nodal period ๐‘‡๐‘‘ , the RAAN-drift ฮ”ฮฉ, and the long term behavior of ๐‘Ž, ๐‘’, and ๐‘– are oscillating at the frequency of the rotation of the argument of periapsis ๐œ”, which has a period of roughly 129 days. 4.2.5 The Bounded Motion Conditions by Xu et al. Considering that each orbit is individually influenced by the gravitational perturbations determining its shape and orbital period, bounded motion conditions link two orbits in space-time. Xu et al. [100] showed that the conditions for bounded motion between two orbits O1 and O2 require the following conditions to be met: ๐‘‡ ๐‘‘ ( O1 ) = ๐‘‡ ๐‘‘ ( O2 ) (4.11) ฮ”ฮฉ ( O1 ) = ฮ”ฮฉ ( O2 ) . (4.12) In other words, any two orbits are in sync, if both, their average nodal period ๐‘‡ ๐‘‘ and their average drift of the ascending node ฮ”ฮฉ, are the same. The time related condition is linked to the space related condition by the space-time event at the ascending node, where the object passes through the equatorial plane from south to north. The time difference between two consecutive ascending nodes is the nodal period ๐‘‡๐‘‘ . The angular difference between two consecutive ascending nodes is denoted by ฮ”ฮฉ, also referred to as the RAAN-drift. It is defined by   ฮ”ฮฉ = ๐œ™ ( ๐‘›+1 ) โˆ’ ๐œ™ ( ๐‘› ) โˆ’ 2๐œ‹sgn ( H๐‘ง ) , (4.13) where โˆ’2๐œ‹sgn ( H๐‘ง ) ensures that ฮ”ฮฉ is the shortest angular distance between the two consecutive ascending nodes. Under zonal perturbation, the nodal period ๐‘‡๐‘‘ and the RAAN-drift ฮ”ฮฉ show regular oscillatory behavior (see Fig. 4.4), making their average values constants of motion. The basic goal of our approach is finding a way of cleverly calculating those average values and relating them to the constants of motion H๐‘ง and ๐ธ. Given the relation, H๐‘ง and ๐ธ can be chosen such that the bounded motion conditions are satisfied and the associated orbits are bound. 69 4.2.6 The Fixed Point Orbit Under zonal perturbation, there are special orbits for which the nodal period ๐‘‡๐‘‘ and the RAAN-drift ฮ”ฮฉ are constant. The associated reduced state Z = (๐œŒ, ๐‘ฃ ๐œŒ , ๐‘ง = 0, ๐‘ฃ ๐‘ง ) at the ascending nodes remains unchanged, which is why these orbits are called fixed point orbits. The orbits are also known as quasi-circular orbits, which originates from the idea of having the elliptical reference shape of the orbit rotate within the orbital plane under zonal perturbation. The fact that ๐‘Ÿ = ๐œŒ is constant at the ascending node for those orbits suggests that the rotating reference shape of the orbit in the orbital plane is a circle. The Keplerian elements of such a quasi-circular orbit (see Fig. 4.2) show however that ๐‘’ oscillates around a value slightly greater than zero, which is the reason for the word โ€˜quasiโ€™. More insightful is the idea that the perturbations influence the orbit just right to yield periodic behavior after just one orbital revolution around the central body. Compared to the Keplerian elements of non-quasi-circular orbits like the one shown in Fig. 4.1, the orbital behavior of the quasi-circular orbit is a lot more regular. Its nodal period ๐‘‡๐‘‘ and ascending node drift ฮ”ฮฉ are constant and not oscillating as Fig. 4.4 reveals. Since the long term oscillation has 1.141 ๐‘Ž [๐‘…0 ] 1.14 1.139 1.138 0.003 ๐‘’ 0.002 0.001 0 ๐œ” [rad] ฮฉ [rad] ๐‘– [rad] 1.72802 1.72795 1.72788 0.003 0.002 0.001 0 2๐œ‹ 0 0 50 100 150 200 time [min] Figure 4.2: Keplerian elements of a quasi-circular low Earth orbit under Earthโ€™s zonal gravitational perturbation. 70 no amplitude, the entire dynamics of a quasi-circular orbit are already captured by the time scale of minutes shown in Fig. 4.2. For our approach, these fixed point orbits serve as a reference for entire families of orbits which all share the same average nodal period ๐‘‡๐‘‘ and the same average RAAN-drift ฮ”ฮฉ. Our method calculates a manifold in (๐œŒ, ๐‘ฃ ๐œŒ , ๐‘ง, ๐‘ฃ ๐‘ง , H๐‘ง , ๐ธ) around the fixed point, where the manifold is defined such that any two points on the manifold satisfy the bounded motion condition. In the fully gravitationally perturbed system the axial symmetry vanishes, which introduces a ๐œ™ dependence and results in H๐‘ง no longer being a constant of motion. Accordingly, fixed point orbits in the fully gravitationally perturbed systems must have a fixed point property in the full state X = (๐œŒ, ๐‘ฃ ๐œŒ , ๐‘ง = 0, ๐‘ฃ ๐‘ง , ๐œ™, H๐‘ง ). We will discuss fixed point orbits in the fully perturbed system and the possibilities of creating bounded motion manifolds around them in more detail later in this chapter, but first, we will present the method and results from [95], where manifolds of bounded motion orbits for the zonal problem are calculated. 4.3 Method of Bounded Motion Design Under Zonal Perturbation This section is from [95]. The goal is to generate a Poincarรฉ return map P that describes the dynamics of the system by characterizing how a state ( Xini , ๐‘ก = 0) โˆˆ O within a Poincarรฉ surface S returns to S. Defining a suitable Poincarรฉ surface is the first step in generating the map. Secondly, a reference orbit with fixed point properties has to be identified to ensure that the expansion point of the map returns to itself. The Poincarรฉ return map is then calculated as an expansion around the reference orbit before being averaged using DA normal form methods. This yields the average nodal period ๐‘‡ ๐‘‘ and average ascending node drift ฮ”ฮฉ as a function of the system parameters and expansion variables around the reference orbit. Using DA inversion methods, the system parameters can be determined such that the bounded motion conditions are met. 71 4.3.1 The Poincarรฉ Surface Space The bounded motion conditions are defined regarding the ascending node of two orbits. To be able to enforce the bounded motion condition on our map, we choose the set of ascending nodes (๐‘ง = 0, ๐‘ฃ ๐‘ง โ‰ฅ 0) as the Poincarรฉ surface. The Poincarรฉ surface S  can be divided into subsurfaces  S ,H๐‘ง ,๐ธ for specific angular momentum components H๐‘ง and energies ๐ธ. These surfaces contain all states with the parameters ( H๐‘ง , ๐ธ) that lie in the equatorial plane (๐‘ง = 0) and satisfy ๐‘ฃ ๐‘ง > 0. The restriction of ๐‘ฃ ๐‘ง to positive values makes the relation between ๐ธ and ๐‘ฃ ๐‘ง (see Eq. (4.8)) bijective and  therefore locally invertible in S ,H๐‘ง ,๐ธ , so โˆš๏ธ„ H๐‘ง 2 ๏ฃด ๏ฃฑ   ๏ฃผ  ๏ฃด ๏ฃฒ ๏ฃด ๏ฃด ๏ฃฝ S ,H๐‘ง ,๐ธ = X | ๐‘ง = 0, ๐‘ฃ ๐‘ง = 2 (๐ธ โˆ’ ๐‘ˆ (๐‘Ÿ)) โˆ’ ๐‘ฃ ๐‘Ÿ2 โˆ’ . (4.14) ๏ฃด ๏ฃด ๐‘Ÿ ๏ฃด ๏ฃด ๏ฃณ ๏ฃพ  This means that any state X โˆˆ S ,H๐‘ง ,๐ธ is uniquely determined by (๐‘Ÿ, ๐‘ฃ ๐‘Ÿ , ๐œ™), since ๐‘ง = 0 and ๐‘ฃ ๐‘ง (๐‘Ÿ, ๐‘ฃ ๐‘Ÿ , H๐‘ง , ๐ธ). 4.3.2 The Fixed Point Orbit The orbit associated with the fixed point state is called reference orbit. The reference orbit has the special property that it returns to the same reduced state Z = (๐‘Ÿ, ๐‘ฃ ๐‘Ÿ , ๐‘ง, ๐‘ฃ ๐‘ง )๐‘‡ after each revolution with a constant nodal period ๐‘‡๐‘‘โ˜… and a constant angle advancement in ๐œ™, which is also referred to as the fixed point drift in the ascending node ฮ”ฮฉโ˜…. For a certain set of parameters ( H๐‘ง , ๐ธ), we use DA inversion techniques iteratively to find the fixed point orbit (see Sec. 2.2 and Sec. 2.3). The iteration is initialized with an educated guess of the fixed point corresponding to the ascending node state  ๐‘‡ 1 Z0 = ๐‘Ÿ = โˆ’ , ๐‘ฃ ๐‘Ÿ = 0, ๐‘ง = 0, ๐‘ฃ ๐‘ง (๐‘Ÿ, H๐‘ง , ๐ธ) , (4.15) 2๐ธ where ๐‘Ÿ is set to the apsides of an elliptical orbit in an unperturbed gravitational field with a specific orbital energy ๐ธ (see [97]). For each iteration step ๐‘›, the state Z๐‘›โˆ’1 is expanded in the variables (๐‘Ÿ, ๐‘ฃ ๐‘Ÿ ). After a full orbit integration until the next ascending node intersection, the map M is timewise projected onto the 72  Poincarรฉ surface S ,H๐‘ง ,๐ธ (see Sec. 2.2). The resulting Poincarรฉ map P represents the one turn map in dependence on variations (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ ) in the variables (๐‘Ÿ, ๐‘ฃ ๐‘Ÿ ). The difference between the constant part of the map P and the initial state Z๐‘›โˆ’1 in the components ๐‘Ÿ and ๐‘ฃ ๐‘Ÿ is denoted by ฮ”๐‘Ÿ and ฮ”๐‘ฃ ๐‘Ÿ , respectively. The Poincarรฉ map without its constant part is indicated by P 0. The next initial state Z๐‘› for the iterative process will be given by the evaluation of โˆ’1 ยฉ Z๐‘Ÿ,๐‘› ยช ยฉ P๐‘Ÿ0 (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ ) โˆ’ ๐›ฟ๐‘Ÿ ยช ยญ ยญ ยฎ=ยญ ยฎ ยญ 0 ยฎ ยฎ (๐›ฟ๐‘Ÿ = โˆ’ฮ”๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ = โˆ’ฮ”๐‘ฃ ๐‘Ÿ ) . (4.16) Z๐‘ฃ๐‘Ÿ ,๐‘› P๐‘ฃ๐‘Ÿ (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ ) โˆ’ ๐›ฟ๐‘ฃ ๐‘Ÿ ยซ ยฌ ยซ ยฌ The process is repeated until the offset (ฮ”๐‘Ÿ, ฮ”๐‘ฃ ๐‘Ÿ ) is smaller than a threshold value e.g. 1E-14. The resulting Z๐‘› is then the ascending node state of the fixed point orbit. 4.3.3 The Calculation of Poincarรฉ Return Map Given a fixed point state Z โ˜… from Sec. 4.3.2 for the parameter set ( H๐‘ง , ๐ธ), the Poincarรฉ return map   P : (S , ๐‘ก) โ†’ (S , ๐‘ก) is calculated as a DA expansion around that reference orbit. In the first step, the flow M of the fixed point and its neighborhood in S  (expansion in (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ๐‘Ÿ , ๐›ฟH๐‘ง , ๐›ฟ๐ธ)) is obtained by integrating the system of ODEs from the initial state until the reference/fixed point orbit  is an element of S ,H๐‘ง ,๐ธ again after ๐‘‡๐‘‘โ˜…. In other words, the state is integrated until the orbit of X 0 intersects with the equatorial plane from south to north again.  While the reference orbit itself is in S ,H๐‘ง ,๐ธ โŠ‚ S  after ๐‘‡๐‘‘โ˜…, the expansion around the reference orbit is not in S,H๐‘ง +๐›ฟH๐‘ง ,๐ธ+๐›ฟ๐ธ โŠ‚ S due to changing nodal periods of the orbits within the expansion. In order to project the flow M after ๐‘‡๐‘‘โ˜… onto the Poincarรฉ surface S,๐ธ+๐›ฟH๐‘ง ,๐ธ+๐›ฟH๐‘ง , a timewise projection is calculated following Sec. 2.2 and [40]. The flow M is expanded in time to find the intersection time ๐‘กintersec (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ , ๐›ฟH๐‘ง , ๐›ฟ๐ธ) such that P๐‘ง = M๐‘ง (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ , ๐›ฟH๐‘ง , ๐›ฟ๐ธ, ๐‘กintersec (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ , ๐›ฟH๐‘ง , ๐›ฟ๐ธ)) = 0 (4.17)   and P = ( M (๐‘กintersec ), ๐‘‡๐‘‘โ˜… + ๐‘กintersec ) โˆˆ (S ,H๐‘ง +๐›ฟH๐‘ง ,๐ธ+๐›ฟ๐ธ , ๐‘ก) โŠ‚ (S , ๐‘ก). The time component P๐‘‡๐‘‘ of the Poincarรฉ return map yields the dependence of the nodal period ๐‘‡๐‘‘ on the system parameters and expansion variables. 73 4.3.4 The Normal Form Averaging Given the fixed point Poincarรฉ return map P with ยฉ P๐‘Ÿ (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ , ๐›ฟH๐‘ง , ๐›ฟ๐ธ) ยช ยญ ยฎ ยญ P๐‘ฃ๐‘Ÿ (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ , ๐›ฟH๐‘ง , ๐›ฟ๐ธ) ยฎ ยญ ยฎ ยญ ยฎ ยญ ยฎ ยญ P๐‘ง = 0 ยฎ P (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ , ๐›ฟH๐‘ง , ๐›ฟ๐ธ) = ยญ (4.18) ยญ ยฎ ยฎ ยญ P ยญ ๐‘ฃ๐‘ง (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ , ๐›ฟ H ๐‘ง , ๐›ฟ๐ธ) ยฎ ยฎ ยญ ยฎ ยญ P๐œ™ (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ , ๐›ฟH๐‘ง , ๐›ฟ๐ธ) ยฎ ยญ ยฎ ยญ ยฎ ยญ ยฎ P๐‘‡๐‘‘ (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ , ๐›ฟH๐‘ง , ๐›ฟ๐ธ) ยซ ยฌ we are using only the first two components (in ๐‘Ÿ and ๐‘ฃ ๐‘Ÿ ) of the Poincare map for the calculation of phase space transformation provided by the DA normal form algorithm, since the motion is determined by only the (๐‘Ÿ, ๐‘ฃ ๐‘Ÿ ) phase space and the parameters ( H๐‘ง , ๐ธ). The reduced map is denoted by K = ( P๐‘Ÿ , P๐‘ฃ๐‘Ÿ )๐‘‡ . The normal form transformation A (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ , ๐›ฟH๐‘ง , ๐›ฟ๐ธ) (see Eq. (2.33)) and its inverse are used to transform the map K such that A โ—ฆ K โ—ฆ Aโˆ’1 (๐‘ž NF , ๐‘ NF , ๐›ฟH๐‘ง , ๐›ฟ๐ธ) = KNF (๐‘ž NF , ๐‘ NF , ๐›ฟH๐‘ง , ๐›ฟ๐ธ) (4.19) is rotational invariant in the normal form phase space coordinates (๐‘ž NF , ๐‘ NF ) up to the order of calculation. In other words, the distorted phase space curves in original phase space coordi- nates ( P๐‘Ÿ (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ , ๐›ฟH๐‘ง , ๐›ฟ๐ธ), P๐‘ฃ๐‘Ÿ (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ , ๐›ฟH๐‘ง , ๐›ฟ๐ธ)) are transformed to circles in the normal form coordinates (๐‘„ NF (๐‘ž NF , ๐‘ NF , ๐›ฟH๐‘ง , ๐›ฟ๐ธ), ๐‘ƒNF (๐‘ž NF , ๐‘ NF , ๐›ฟH๐‘ง , ๐›ฟ๐ธ)) as Fig. 4.3 illustrates. By rewriting the normal form coordinates (๐‘ž NF , ๐‘ NF ) in an action-angle representation (๐‘Ÿ NF , ฮ›) with ยฉ ๐‘ž NF ยช ยฉcos ฮ›ยช ยญ ยญ ยฎ = ๐‘Ÿ NF ยญ ยฎ ยญ ยฎ, ยฎ (4.20) ๐‘ sin ฮ› ยซ NF ยฌ ยซ ยฌ each normal form phase space curve is characterized by the normal form radius (action) ๐‘Ÿ NF and the path along each curve is parameterized by the angle ฮ›. Using the inverse normal form transformation Aโˆ’1 (see Eq. (2.34)), the original phase space variables (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ ) of P (and K) are expressed in 74 a) b) 1 1 ๐‘Ÿ๐‘– ๐‘Ÿ NF 0.5 0.5 ๐‘ 0 ๐‘ NF 0 โˆ’0.5 โˆ’0.5 โˆ’1 โˆ’1 ฮ›๐‘˜ ฮ› โˆ’1 โˆ’0.5 0 0.5 1 โˆ’1 โˆ’0.5 0 0.5 1 ๐‘ž ๐‘ž NF Figure 4.3: a) Distorted phase space behavior in the original phase space (๐‘ž, ๐‘) and b) circular behavior in the corresponding normal form phase space (๐‘ž NF , ๐‘ NF ). In a), the phase space angle advancement ฮ› ๐‘˜ and the phase space radius ๐‘Ÿ๐‘– are not constant by continuously change along each of the phase space curves. In b), the phase space behavior is rotationally invariant (โ€˜normalizedโ€™) with a constant radius ๐‘Ÿ NF and a constant but amplitude dependent angle advancement ฮ›(๐‘Ÿ NF ). terms of the action-angle representation and variations in the system parameters (๐›ฟH๐‘ง , ๐›ฟ๐ธ): (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ ) = Aโˆ’1 (๐‘ž NF (๐‘Ÿ NF , ฮ›) , ๐‘ NF (๐‘Ÿ NF , ฮ›) , ๐›ฟH๐‘ง , ๐›ฟ๐ธ) . (4.21) The Poincarรฉ map P (๐‘Ÿ NF , ฮ›, ๐›ฟH๐‘ง , ๐›ฟ๐ธ) is then averaged over a full phase space revolution, by integrating along the angle ฮ›: โˆฎ 1 P (๐‘Ÿ NF , ๐›ฟH๐‘ง , ๐›ฟ๐ธ) = P (๐‘Ÿ NF , ฮ›, ๐›ฟH๐‘ง , ๐›ฟ๐ธ) ๐‘‘ฮ›. (4.22) 2๐œ‹ The numerical averaging presented in [42] is done in the time domain, which cannot incorporate the slightly different oscillation frequencies of the relevant quantities ๐‘‡๐‘‘ and ฮ”ฮฉ for the different orbits. The key advantage of the normal form representation is that the different oscillation frequencies are captured by the amplitude dependent angle advancement in the normal form. The generalized parameterization of all normal form phase space curves makes the averaging independent of those differences in the frequency. Splitting the integration into subsections minimizes the error of the numerical integration and considerably improves the quality and accuracy of the averaging. For ๐‘› separate parameterization     2๐œ‹(๐‘˜โˆ’1) 2๐œ‹(๐‘˜โˆ’1) ยฉ๐‘ž NF ยช ยฉcos โˆ’ sin ยช ยฉcos ฮ›ยช ยญ ยฎ = ๐‘Ÿ NF ยญ  ๐‘›   ๐‘›  ยฎยญ ยฎ ๐‘˜ โˆˆ {1, 2, ..., ๐‘›} , (4.23) ยญ ยฎ ยญ 2๐œ‹(๐‘˜โˆ’1) 2๐œ‹(๐‘˜โˆ’1) ยฎ ยญ ยฎ ๐‘ž NF sin ๐‘› cos ๐‘› sin ฮ› ยซ ยฌ ยซ ยฌยซ ยฌ 75 each section is integrated over the symmetric interval of ฮ› โˆˆ [โˆ’๐œ‹/๐‘›, ๐œ‹/๐‘›]. The result of the averaging yields every component of P averaged over a full phase space curve. In particular, it yields the averaged drift in the ascending node ฮ”ฮฉ (๐‘Ÿ NF , ๐›ฟH๐‘ง , ๐›ฟ๐ธ) and average nodal period ๐‘‡ ๐‘‘ (๐‘Ÿ NF , ๐›ฟH๐‘ง , ๐›ฟ๐ธ). For mission design purposes the abstract quantity ๐‘Ÿ NF is expressed by the original coordinates (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ ) and the parameters (๐›ฟH๐‘ง , ๐›ฟ๐ธ) with   2 (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ , ๐›ฟ H , ๐›ฟ๐ธ) ๐‘Ÿ NF = ๐‘ž 2NF + ๐‘ 2NF (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ , ๐›ฟH๐‘ง , ๐›ฟ๐ธ) (4.24) ๐‘Ÿ ๐‘ง using the normal form transformation A, which yields how (๐‘ž NF , ๐‘ NF ) depend on the original coordinates (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ ) and the parameters (๐›ฟH๐‘ง , ๐›ฟ๐ธ). The average drift in the ascending node ฮ”ฮฉ (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ , ๐›ฟH๐‘ง , ๐›ฟ๐ธ) and the average nodal period ๐‘‡ ๐‘‘ (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ , ๐›ฟH๐‘ง , ๐›ฟ๐ธ) are then projected such that the bounded motion conditions are satisfied, with ฮ”ฮฉโ˜… = ฮ”ฮฉ (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ , ๐›ฟH๐‘ง (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ ) , ๐›ฟ๐ธ (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ )) (4.25) ๐‘‡๐‘‘โ˜… = ๐‘‡ ๐‘‘ (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ , ๐›ฟH๐‘ง (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ ) , ๐›ฟ๐ธ (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ )) . (4.26) In this process, DA inversion methods are used to find ๐›ฟH๐‘ง (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ ) and ๐›ฟ๐ธ (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ ). The dependence of H๐‘ง and ๐ธ on orbital parameters for bounded motion orbits were previously discussed in [90, 79]. Theoretically, one could have proceeded with the abstract invariant of motion ๐‘Ÿ NF to satisfy the bounded motion condition with ๐›ฟH๐‘ง (๐‘Ÿ NF ) and ๐›ฟ๐ธ (๐‘Ÿ NF ). For specific bounded orbits one would then have chosen a value for ๐‘Ÿ NF to calculate (๐›ฟH๐‘ง , ๐›ฟ๐ธ) and afterwards the initial values for (๐‘Ÿ, ๐‘ฃ ๐‘Ÿ ) by using Eq. (4.21), where ฮ› can be chosen freely. 4.4 Bounded Motion Results The following results were already discussed in [95]. In this section, we will apply the normal form methods for bounded motion of low Earth and medium Earth orbits. For this, we use fixed point orbits of the zonal problem that have previously been investigated by He et al. [42] for the low Earth orbit (LEO) and Baresi and Scheeres [10] for the medium Earth orbit (MEO). 76 As explained above, the fixed point Poincarรฉ maps P are calculated as an expansion in the variables (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ , ๐›ฟH๐‘ง , ๐›ฟ๐ธ) around the respective fixed point orbit. In the calculation we consider zonal perturbations up to the ๐ฝ15 -term, since investigations in [42] indicated no considerable influence of ๐ฝ ๐‘˜ terms for ๐‘˜ > 15. We are using DA maps of 8th order, which provide the best balance of accuracy and computation time. Additionally, the following dimensionless units are used: distances are considered in units of the average Earth radius ๐‘…0 = 6378.137 km and time is considered in units of ๐‘‡0 = 806.811 s such that the gravitational constant assumes the value ๐œ‡ = 1. Thus, velocities are in units of ๐‘…0 /๐‘‡0 = 7.905 km/s. It will be shown that the DANF method provides entire sets of bounded motions that extend far beyond the realistic/practical scope. Since the approach is based on polynomial expansions, it is obvious it will have to fail at some point. After presenting the bounded motion results for the LEO and MEO case, we take a look at the limitations of the DANF method and the resulting sets for very large distances between orbits. 4.4.1 Bounded Motion in Low Earth Orbit In a first comparison, we are investigating bounded motion around a pseudo-circular LEO that was also considered in [42]. The pseudo-circular orbit corresponds to the reduced fixed point state (๐‘Ÿ โ˜…, ๐‘ฃโ˜… ๐‘Ÿ ) = (1.14016749, โˆ’1.05621369E-3) (4.27) for the parameters ( H๐‘ง , ๐ธ) = (โˆ’0.16707295, โˆ’0.43870527). The orbit has a fixed nodal period of ๐‘‡๐‘‘โ˜… = 7.64916169 (โ‰ˆ 103 min) and a constant ascending node drift of ฮ”ฮฉโ˜… = 1.22871195E-3 rad (0.0704โ—ฆ ). The vertical position ๐‘ง of the Poincarรฉ fixed point orbit are defined by the Poincarรฉ  section (๐‘ง = 0) and Eq. (4.14) with ๐‘ฃโ˜… ๐‘ง ๐‘Ÿ , ๐‘ฃ ๐‘Ÿ , H๐‘ง , ๐ธ = 0.92518953. โ˜… โ˜… The computation of the Poincarรฉ map took 165 seconds on a Lenovo E470 with an IntelยฎCoreTM i5-7200U CPU 2.5GHz. The map confirms the fixed point property of the orbit, since the offset of the constant part of the map from the initial coordinates is well within the numerical error of the integration with (ฮ”๐‘Ÿ, ฮ”๐‘ฃ ๐‘Ÿ , ฮ”๐‘ง, ฮ”๐‘ฃ ๐‘ง ) = (4E-15, 5E-13, โˆ’1E-15, โˆ’4E-15). The normal form 77 transformation of the reduced fixed point Poincarรฉ map K = ( P๐‘Ÿ , P๐‘ฃ๐‘Ÿ )๐‘‡ is calculated via the DA normal form algorithm (in 90 milliseconds). The circular phase space behavior in normal form space is parameterized using the action-angle notation (๐‘Ÿ NF , ฮ›). The phase space parameterization is transformed back to the original coordinates of the Poincarรฉ map. The Poincarรฉ map is averaged (in 52 milliseconds) over a full phase space rotation using 8 subsections following the procedure outlined in Sec. 4.3.4. Afterwards, the variable ๐‘Ÿ NF is expressed in terms of ๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ , ๐›ฟH๐‘ง and ๐›ฟ๐ธ before the variations in the constants of motion (๐›ฟH๐‘ง , ๐›ฟ๐ธ) are matched dependent on (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ ) such that the averaged expressions for ๐‘‡๐‘‘ and ฮ”ฮฉ satisfy the bounded motion conditions (Eq. (4.25) and Eq. (4.26)). Note that above we are not listing the computation time for the computation steps that are performed very quickly. Considering bounded orbits initiated with the same ๐‘ฃ ๐‘Ÿ as the pseudo-circular orbit (๐›ฟ๐‘ฃ ๐‘Ÿ = 0), the dependence of H๐‘ง (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ = 0) and ๐ธ (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ = 0) are provided in Tab. 4.1. Table 4.1: The expansion of H๐‘ง (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ = 0) and ๐ธ (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ = 0) for relative bounded motion orbits with an average nodal period ๐‘‡๐‘‘ = 7.64916169 (โ‰ˆ103 min) and an average ascending node drift of ฮ”ฮฉ = 1.22871195E-3 rad. The expansion is relative to the pseudo-circular LEO from [42]. H๐‘ง (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ = 0) = ๐ธ (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ = 0) = โˆ’ 0.16707295 โˆ’ 0.43870527 + 0.32072807 ๐›ฟ๐‘Ÿ 2 โˆ’ 0.31602983E-3 ๐›ฟ๐‘Ÿ 2 + 0.25767948E-3 ๐›ฟ๐‘Ÿ 3 โˆ’ 0.25390482E-6 ๐›ฟ๐‘Ÿ 3 โˆ’ 0.19132824 ๐›ฟ๐‘Ÿ 4 โˆ’ 0.31003174E-3 ๐›ฟ๐‘Ÿ 4 + 0.53296708E-4 ๐›ฟ๐‘Ÿ 5 โˆ’ 0.85361819E-6 ๐›ฟ๐‘Ÿ 5 + 0.12006391E-1 ๐›ฟ๐‘Ÿ 6 โˆ’ 0.32152252E-3 ๐›ฟ๐‘Ÿ 6 + 0.60713391E-3 ๐›ฟ๐‘Ÿ 7 โˆ’ 0.24661573E-5 ๐›ฟ๐‘Ÿ 7 โˆ’ 0.19751494 ๐›ฟ๐‘Ÿ 8 โˆ’ 0.21784073E-3 ๐›ฟ๐‘Ÿ 8 To show that the expansion of ๐›ฟH๐‘ง and ๐›ฟ๐ธ provide relative bounded motion orbits, we illustrate the long term behavior of three LEOs relative to one another. The first orbit is the fixed point/pseudo- circular orbit and is denoted by O0 . The second orbit (O1 ) is initiated at ๐›ฟ๐‘Ÿ = 0.06 with ๐›ฟ๐‘ฃ ๐‘Ÿ = 0. The third orbit (O2 ) is initiated at ๐›ฟ๐‘Ÿ = 0.13 with ๐›ฟ๐‘ฃ ๐‘Ÿ = 0. The last two both have an initial longitudinal offset of ๐œ™ = 0.5โ—ฆ relative to O0 . The specific values of the orbits are given in Tab. 4.2. In the (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ ) phase space, O1 and O2 oscillate around the fixed point (๐‘Ÿ 0 , ๐‘ฃ ๐‘Ÿ,0 ) of O0 . 78 Table 4.2: The LEOs below are all initiated at ๐‘ฃ ๐‘Ÿ,0 = โˆ’1.05621369E-3 and ๐‘Ÿ 0 = 1.14016749 + ๐›ฟ๐‘Ÿ, and have an average nodal period of ๐‘‡๐‘‘ = 7.64916169 (โ‰ˆ103 min) and an average ascending node drift of ฮ”ฮฉ = 1.22871195E-3 rad. The pseudo-circular LEO from [42] is denoted by O0 . ๐›ฟ๐‘Ÿ ๐›ฟ๐‘ฃ ๐‘Ÿ ๐œ™ H๐‘ง ๐ธ O0 0.00 0 0.0โ—ฆ -0.16707295 -0.43870527 O1 0.06 (383 km) 0 0.5โ—ฆ -0.16592075 -0.43870642 O2 0.13 (829 km) 0 0.5โ—ฆ -0.16170668 -0.43871071 Accordingly, the altitude of those orbits is also changing and roughly captured by ๐‘Ÿ 0 ยฑ ๐›ฟ๐‘Ÿ, which means that O2 already reaches very low altitudes around ๐‘Ÿ = 1.01. In Fig. 4.4 we show that the bounded motion conditions are met: the oscillatory behavior of the nodal period ๐‘‡๐‘‘ and the ascending node drift ฮ”ฮฉ of the two orbits O1 and O2 average out to the same value, respectively, which corresponds to the constant nodal period ๐‘‡๐‘‘โ˜… and constant ascending node drift ฮ”ฮฉโ˜… of the fixed point orbit O0 . 7.6503 O0 7.6496 O1 ๐‘‡๐‘‘ [-] 7.6489 O2 7.6482 0.07044 ฮ”ฮฉ [deg] 0.07041 0.07038 0.07035 0 1000 2000 3000 4000 5000 6000 7000 orbital revolutions 0 50 100 150 200 250 300 350 400 450 500 approx. time [days] Figure 4.4: Oscillatory behavior of the bounded motion quantities ๐‘‡๐‘‘ and ฮ”ฮฉ of the bounded LEOs O1 and O2 initiated at ๐›ฟ๐‘Ÿ = 0.06 and ๐›ฟ๐‘Ÿ = 0.13, respectively. Additionally, the constant nodal period ๐‘‡๐‘‘โ˜… = 7.64916169 and constant ascending node drift of ฮ”ฮฉโ˜… = 0.0704โ—ฆ of the fixed point orbit O0 are shown. The periods of oscillation are 1763 orbital revolutions (126 days) for O2 , 1810 orbital revolutions (129 days) for O1 , and 1823 orbital revolutions (130 days) for ๐›ฟ๐‘Ÿ โ†’ 0 of O0 . The shown results are generated by numerical integration. The bounded motion is further confirmed by Fig. 4.5, which shows the total distance between the three LEOs respectively for 14 years. Furthermore, Fig. 4.5 illustrates the relative radial and along-track distance between the orbit pairs from the perspective of one of the orbits in the pair. 79 2.5 O1 from O0 O2 from O0 2 O2 from O1 along-track distance [1000 km] relative distance [1000 km] 2 1 1.5 0 1 โˆ’1 0.5 โˆ’2 0 2 4 6 8 10 12 14 โˆ’1 โˆ’0.5 0 0.5 1 time [years] radial distance [1000 km] Figure 4.5: Relative bounded motion of LEOs with an average nodal period of ๐‘‡๐‘‘ = 7.64916169 (โ‰ˆ103 min) and an average node drift of ฮ”ฮฉ = 1.22871195E-3 rad for 14 years. The total relative distance between the orbits is shown in the left plot and the right plot shows the relative radial and along-track distance between orbit pairs from the perspective of one of the orbits in the pair. The oscillation in the relative distance between O2 and O1 is caused by the rotating orbital orientation of the orbits at different frequencies. Apart from yielding long term bounded motion, the normal form methods also provide the average angle advancement ฮ› in the (๐‘Ÿ, ๐‘ฃ ๐‘Ÿ ) phase space. This angle advancement is directly linked to the rotation frequency ๐œ” ๐‘ of the orbit (and its apsides) within its orbital plane, which causes the oscillation of ๐‘‡๐‘‘ and ฮ”ฮฉ shown in Fig. 4.4 with ๐œ” ๐‘ . One (๐‘Ÿ, ๐‘ฃ ๐‘Ÿ ) phase space rotation corresponds to one revolution of the orbit (and its apsides) within its orbital plane. Accordingly, the frequency ๐œ” ๐‘ = ฮ›/2๐œ‹ is equivalent to the definition of the tune and the tune shifts ๐œˆ + ๐›ฟ๐œˆ, which are just the normalized angle advancement separated into its constant part (the tune ๐œˆ) and its amplitude dependent part (the tune shifts ๐›ฟ๐œˆ). The normal form yields the average angle advancement ฮ› dependent on (๐‘Ÿ NF , ๐›ฟH๐‘ง , ๐›ฟ๐ธ). After normalizing ฮ›, by division by 2๐œ‹, and replacing ๐‘Ÿ NF by an expression of (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ ) and (๐›ฟH๐‘ง , ๐›ฟ๐ธ) according to Eq. (4.24), and using the expressions for (๐›ฟH๐‘ง (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ ), ๐›ฟ๐ธ (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ )) from earlier, the frequency ๐œ” ๐‘ (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ ) is obtained for the bounded motion orbits around the fixed point LEO. The coefficients of ๐œ” ๐‘ for ๐›ฟ๐‘ฃ ๐‘Ÿ = 0 are given in Tab. 4.3. 80 Table 4.3: Expansion of ๐œ” ๐‘ (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ = 0) of relative bounded motion LEOs with an average nodal period ๐‘‡๐‘‘ = 7.64916169 (โ‰ˆ103 min) and an average node drift of ฮ”ฮฉ = 1.22871195E-3 rad. The expansion is relative to the pseudo-circular LEO from [42]. ๐œ” ๐‘ (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ = 0) = + 0.54868728E-3 + 0.10803872E-2 ๐›ฟ๐‘Ÿ 2 + 0.86800515E-6 ๐›ฟ๐‘Ÿ 3 + 0.10552068E-2 ๐›ฟ๐‘Ÿ 4 + 0.29106874E-5 ๐›ฟ๐‘Ÿ 5 โˆ’ 0.76284414E-3 ๐›ฟ๐‘Ÿ 6 + 0.39324207E-5 ๐›ฟ๐‘Ÿ 7 โˆ’ 0.35077526E-1 ๐›ฟ๐‘Ÿ 8 Accordingly, the periods of the oscillations of the nodal periods ๐‘‡๐‘‘ and the ascending node drifts ฮ”ฮฉ in Fig. 4.4 (in units of orbital revolutions) are just the inverse of the frequencies ๐œ” ๐‘ (๐›ฟ๐‘Ÿ = 0.06) = 5.52590498E-4 and ๐œ” ๐‘ (๐›ฟ๐‘Ÿ = 0.13) = 5.67242676E-4. These frequencies also help explain the oscillation of the total relative distance range between O1  O2 over 13.3 years in Fig. 4.5. While O1 shows repetitive behavior after 1809.7 orbital revolutions (129.3 days), the behavior of O2 is repetitive after 1762.9 orbital revolutions (125.9 days). Accordingly, the two orbits will be in and out of sync regarding their orbital orientation, while maintaining bounded due to the matching average nodal period and ascending node drift. Specifically, the two orbits will be back in sync after about 68170 orbital revolutions (4869 days โ‰ก 13.3 years) as Fig. 4.5 illustrates, since O1 will have turned 37.7 times while O2 will have turn exactly once less, namely, 36.7 times, bringing them both back into the same orbital orientation to one other before moving apart again. In conclusion, our first comparison showed the superiority of the normal form methods, particularly, compared to the iterative map evaluation method in [42], where numerical adjustments to the method were required to provide long term relative bounded motion for ๐›ฟ๐‘Ÿ = 0.11. In Sec. 4.4.3 we will show that the DANF method even provides hypothetical long term bounded motion up to ๐›ฟ๐‘Ÿ = 0.3, which covers all realistic cases until ๐›ฟ๐‘Ÿ โ‰ˆ 0.14 and further hypothetical (non-practical) cases with altitudes below the Earthโ€™s surface. 81 In the next comparison, we are going to investigate bounded motion much farther from the Earthโ€™s surface. Accordingly, we expect a larger theoretical and practical bounded motion range from the DANF method, due to a weaker influence of the zonal perturbations. 4.4.2 Bounded Motion in Medium Earth Orbit In this comparison, we are considering a medium Earth orbit (MEO) from [10, p. 11] ini- tiated at ๐‘Ÿ = 26562.58 km, ๐‘ฃ ๐‘Ÿ = โˆ’9.05E-4 km/s and ๐‘ฃ ๐‘ง = 3.18 km/s. In the units of ๐‘…0 = 6378.137 km and ๐‘‡0 = 806.811 s, the zonal problem with ๐ฝ2 to ๐ฝ15 yields a fixed point orbit at (๐‘Ÿ โ˜…, ๐‘ฃโ˜… โ˜… ๐‘Ÿ ) = (4.17198963, โˆ’1.14150072E-4) and ๐‘ฃ ๐‘ง = 0.40154964 for the param- eters ( H๐‘ง , ๐ธ) = (1.16863390, โˆ’0.11984818). The fixed point orbit has a fixed nodal period ๐‘‡๐‘‘โ˜… = 53.5395648 (โ‰ˆ12 hours) and constant drift in the ascending node of ฮ”ฮฉโ˜… = โˆ’3.35410945E-4 rad (-0.0192โ—ฆ ). The angular momentum component H๐‘ง is positive for this orbit in contrast to the LEO from Sec. 4.4.1, which means that ๐œ™ยค is positive and the orbit is moving eastwards. The same computer system as in Sec. 4.4.1, took 131 seconds for the computation of the map. The offset of the integration with (ฮ”๐‘Ÿ, ฮ”๐‘ฃ ๐‘Ÿ , ฮ”๐‘ง, ฮ”๐‘ฃ ๐‘ง ) = (โˆ’4E-15, โˆ’2E-13, โˆ’4E-15, 2E-16) is well within the range of the numerical error of the integration. After the normal form transformation (in 100 milliseconds) and the averaging (in 62 milliseconds) following the same procedure as in Sec. 4.4.1, the dependencies of the constants of motion ( H๐‘ง , ๐ธ) on (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ ) were calculated. Below, Tab. 4.4 yields H๐‘ง (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ = 0) and ๐ธ (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ = 0). To illustrate that the DANF methods also provide bounded motion for this set of parameters, we consider the long term behavior of three MEOs relative to one another. The fixed point/pseudo- circular orbit denoted by O0 . Since ๐‘Ÿ โ˜… of the fixed point MEO is about four times the ๐‘Ÿ โ˜… of the low Earth fixed point orbit from the previous section, the bounded orbits are initiated at four times the distance compared to the LEO investigation in Sec. 4.4.1. The orbit O1 is initiated at ๐›ฟ๐‘Ÿ = 0.24 (1531 km) with ๐›ฟ๐‘ฃ ๐‘Ÿ = 0 and O2 is initiated at ๐›ฟ๐‘Ÿ = 0.52 (3317 km) with ๐›ฟ๐‘ฃ ๐‘Ÿ = 0. These relative distances are already larger than distances that are used in practice. Again, both orbits have an initial longitudinal offset of ๐œ™ = 0.5โ—ฆ relative to O0 . The specific values of the orbits are given in Tab. 4.5. 82 Table 4.4: The expansion of H๐‘ง (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ = 0) and ๐ธ (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ = 0) for relative bounded motion MEOs with an average nodal period of ๐‘‡๐‘‘ = 53.5395648 (โ‰ˆ 12 h) and an average ascending node drift of ฮ”ฮฉ = โˆ’3.35410945E-4 rad. The expansion is relative to the pseudo-circular MEO from [10]. H๐‘ง (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ = 0) = ๐ธ (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ = 0) = + 1.16863390 โˆ’ 0.11984818 โˆ’ 0.16787983 ๐›ฟ๐‘Ÿ 2 โˆ’ 0.11295792E-05 ๐›ฟ๐‘Ÿ 2 โˆ’ 0.57819536E-5 ๐›ฟ๐‘Ÿ 3 โˆ’ 0.38903865E-10 ๐›ฟ๐‘Ÿ 3 + 0.72342680E-2 ๐›ฟ๐‘Ÿ 4 โˆ’ 0.16786161E-07 ๐›ฟ๐‘Ÿ 4 + 0.16208617E-6 ๐›ฟ๐‘Ÿ 5 โˆ’ 0.34176382E-11 ๐›ฟ๐‘Ÿ 5 โˆ’ 0.69493130E-4 ๐›ฟ๐‘Ÿ 6 โˆ’ 0.28279909E-08 ๐›ฟ๐‘Ÿ 6 + 0.11561378E-6 ๐›ฟ๐‘Ÿ 7 + 0.27190622E-12 ๐›ฟ๐‘Ÿ 7 + 0.54888817E-4 ๐›ฟ๐‘Ÿ 8 โˆ’ 0.51224108E-10 ๐›ฟ๐‘Ÿ 8 Table 4.5: The MEOs below are all initiated at ๐‘ฃ ๐‘Ÿ,0 = โˆ’1.14150072E-4 and ๐‘Ÿ 0 = 4.17198963 + ๐›ฟ๐‘Ÿ, and have an average nodal period of ๐‘‡ ๐‘‘ = 53.5395648 (โ‰ˆ 12 h) and an average ascending node drift of ฮ”ฮฉ = โˆ’3.35410945E-4 rad. The orbit O0 is the pseudo-circular MEO from [10]. ๐›ฟ๐‘Ÿ ๐›ฟ๐‘ฃ ๐‘Ÿ ๐œ™ H๐‘ง ๐ธ O0 0.0 0 0.0โ—ฆ 1.16863390 -0.119848175 O1 0.24 (1531 km) 0 0.5โ—ฆ 1.15898794 -0.119848240 O2 0.52 (3317 km) 0 0.5โ—ฆ 1.123766254 -0.119848482 Equivalent to Fig. 4.4 we show that the bounded motion conditions are met for the chosen MEOs in Fig. 4.6. The oscillatory behavior of the nodal period ๐‘‡๐‘‘ and the ascending node drift ฮ”ฮฉ of the two orbits O1 and O2 average out to the same value, respectively, which correspond to the constant nodal period ๐‘‡๐‘‘โ˜… and constant ascending node drift ฮ”ฮฉโ˜… of the fixed point orbit O0 . In contrast to the investigated LEOs, the oscillation period of the bounded motion quantities of the MEOs increases with increasing ๐›ฟ๐‘Ÿ. The period of oscillation in the MEO cases is also about two orders of magnitude longer with periods of 47 and 53 years for O1 and O2 , respectively, compared to the LEOs. Using the normal form methods, the rotation frequency ๐œ” ๐‘ of the orbital orientation within its orbital plane is calculated as described in Sec. 4.4.1. The results from the expansion of ๐œ” ๐‘ confirm these periods of oscillation with ๐œ” ๐‘ (0.24) = 2.88842404E-5 and ๐œ” ๐‘ (0.52) = 2.58516089E-5. The expansion of ๐œ” ๐‘ dependent on ๐›ฟ๐‘Ÿ is given in Tab. 4.6. 83 53.5398 O0 O1 ๐‘‡๐‘‘ [-] 53.5396 O2 53.5394 53.5392 -0.0192174 ฮ”ฮฉ [deg] -0.0192176 -0.0192178 -0.0192180 0 10000 20000 30000 40000 50000 60000 70000 80000 orbital revolutions 0 20 40 60 80 100 approx. time [years] Figure 4.6: Oscillatory behavior of the bounded motion quantities ๐‘‡๐‘‘ and ฮ”ฮฉ of the bounded MEOs O1 and O2 initiated at ๐›ฟ๐‘Ÿ = 0.24 and ๐›ฟ๐‘Ÿ = 0.52, respectively. Additionally, the constant nodal period ๐‘‡๐‘‘โ˜… = 53.5395648 and constant ascending node drift of ฮ”ฮฉโ˜… = โˆ’0.0192176316 deg of the fixed point orbit O0 are shown. The periods of oscillation are 38682 orbital revolutions (52.9 years) for O2 , 34621 orbital revolutions (47.4 years) for O1 , and 33671 orbital revolutions (46.1 years) for ๐›ฟ๐‘Ÿ โ†’ 0 of O0 . The shown results are generated by numerical integration. Table 4.6: Expansion of ๐œ” ๐‘ (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ = 0) of relative bounded motion orbits with an average nodal period of ๐‘‡ ๐‘‘ = 53.5395648 (โ‰ˆ 12 h) and an average ascending node drift of ฮ”ฮฉ = โˆ’3.35410945E-4 rad. The expansion is relative to the pseudo-circular MEO from [10]. ๐œ” ๐‘ (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ = 0) = + 0.29699500E-04 โˆ’ 0.14137545E-04 ๐›ฟ๐‘Ÿ 2 โˆ’ 0.48691156E-09 ๐›ฟ๐‘Ÿ 3 โˆ’ 0.22644327E-06 ๐›ฟ๐‘Ÿ 4 โˆ’ 0.43912160E-10 ๐›ฟ๐‘Ÿ 5 โˆ’ 0.10717280E-05 ๐›ฟ๐‘Ÿ 6 โˆ’ 0.10374073E-09 ๐›ฟ๐‘Ÿ 7 + 0.23789772E-05 ๐›ฟ๐‘Ÿ 8 Fig. 4.7 shows the long term bounded motion behavior by illustrating the relative total distance between the orbits and their relative radial and along-track distances. Due to the long oscillation periods in the bounded motion quantities of 47 and 53 years for O1 and O2 , respectively, the oscillation in the total distance between O1 and O2 is about 456 years and can therefore only be partially shown. After 456 years the orbital orientation of O1 will have turned 9.6 times and align again with the orbital orientation of O2 , which will have turned 8.6 times. 84 7 O1 from O0 6 O2 from O0 O2 from O1 along-track distance [1000 km] 6 4 relative distance [1000 km] 5 2 4 0 -2 3 -4 2 -6 0 10 20 30 40 50 60 0 10 20 30 40 50 60 โˆ’3 โˆ’2 โˆ’1 0 1 2 3 time [years] time [years] radial distance [1000 km] Figure 4.7: Relative bounded motion of MEOs from Tab. 4.5 with an average nodal period of ๐‘‡๐‘‘ = 53.5395648 (โ‰ˆ 12 h) and an average ascending node drift of ฮ”ฮฉ = โˆ’3.35410945E-4 rad over 70 years. The total relative distance between the orbits is shown in the left plots and the right plot shows the relative radial and along-track distance between orbit pairs from the perspective of one of the orbits in the pair. The โ€˜breathingโ€™ of the relative total distance between O2 and O0 originates from the rotating orbital orientation of pseudo-elliptical O2 relative to the pseudo-circular O0 . Due to the very long rotation periods, only the first 70 years of the relative distance oscillation and radial/along-track behavior between O2 and O1 could be shown. The โ€˜breathingโ€™ of the relative distance between the orbits is particularly noticeable for the orbit pair of O2 and O0 . The frequency of the โ€˜breathingโ€™ is 2๐œ” ๐‘ which is a result of the rotation of the orbital orientation of the pseudo-elliptical O2 relative to the pseudo-circular O0 . Since the orbital shape of the pseudo-elliptical O2 is approximately symmetric along its semi-major axis, one full rotation of the orbital orientation corresponds to two breathing cycles. In conclusion, our methods also provided an entire set of long term relative bounded motion around the considered fixed point MEO from [10], which was validated far beyond practical relative distances. In the following section, the limitations of our method are investigated. The investigations will show that the validity of the sets presented in Sec. 4.4.1 and Sec. 4.4.2 extends over about twice the already presented distance from their respective fixed point orbits. 85 4.4.3 Testing the Limitations of the DANF Method The previous two sections illustrated the validity of the DANF method for all practical relative distances for bounded motion and beyond. In this section, we move even further away from any practical relevance of the calculated sets of bounded motion to the limitations of our method. Since it is based on polynomial expansions, it is obvious it will fail at some point and we want to show when and how this failing process takes place. First we pick a number of test orbits from the calculated bounded motion sets (see Tab. 4.7). In contrast to previous examples, no initial longitudinal offset relative to the respective fixed point orbits are set. Table 4.7: The following orbit parameters are obtained by evaluating H๐‘ง (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ = 0) and ๐ธ (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ = 0) from Tab. 4.1 and Tab. 4.4 for various ๐›ฟ๐‘Ÿ keeping ๐›ฟ๐‘ฃ ๐‘Ÿ = 0. Test LEOs Test MEOs ๐›ฟ๐‘Ÿ H๐‘ง ๐ธ ๐›ฟ๐‘Ÿ H๐‘ง ๐ธ O0 0.00 -0.16707295 -0.43870527 O0 0.0 1.1686339 -0.11984817 O0.15 0.15 -0.15995246 -0.43871254 O0.6 0.6 1.1091311 -0.11984854 O0.20 0.20 -0.15454760 -0.43871843 O0.7 0.7 1.0881027 -0.11984873 O0.25 0.25 -0.14777078 -0.43872632 O0.8 0.8 1.0641420 -0.11984890 O0.30 0.30 -0.13975416 -0.43873648 O0.9 0.9 1.0373802 -0.11984910 O0.35 0.35 -0.13066556 -0.43874929 O1.0 1.0 1.0079682 -0.11984932 O0.40 0.40 -0.12071669 -0.43876526 O1.1 1.1 0.97607833 -0.11984957 O1.2 1.2 0.94190725 -0.11984984 O1.3 1.3 0.90567972 -0.11985014 O1.4 1.4 0.86765361 -0.11985047 Fig. 4.8 illustrates the behavior of the bounded motion quantities ๐‘‡๐‘‘ and ฮ”ฮฉ for the chosen orbits of the LEO bounded motion set. Both quantities show oscillatory behavior centered at or close to ๐‘‡๐‘‘โ˜… and ฮ”ฮฉโ˜…, respectively. With increasing distance ๐›ฟ๐‘Ÿ, the influence of higher order oscillations becomes apparent. The frequency and amplitude of oscillation of the bounded motion quantities also increase with increasing distance ๐›ฟ๐‘Ÿ. If the bounded motion conditions are not met or only met approximately, the orbits will start drifting apart. This effect is illustrated in Fig. 4.9, which shows very slowly diverging behavior of approximately 2.6 km/year for ๐›ฟ๐‘Ÿ = 0.3 (1913 km) and a stronger divergence of approximately 10.6 86 7.654 7.652 O0 ๐‘‡๐‘‘ [-] 7.65 O0.15 7.648 O0.20 O0.25 7.646 O0.30 0.0708 O0.35 0.0706 O0.40 ฮ”ฮฉ [deg] 0.0704 0.0702 0.07 0.0698 0 500 1000 1500 2000 2500 3000 orbital revolutions 0 50 100 150 200 approx. time [days] Figure 4.8: The behavior of the bounded motion quantities ๐‘‡๐‘‘ and ฮ”ฮฉ for the test orbits from Tab. 4.7 of the calculated LEO bounded motion set generated by numerical integration. For large ๐›ฟ๐‘Ÿ, the influences of higher order oscillations are apparent. The frequency and amplitude of oscillation increase with increasing ๐›ฟ๐‘Ÿ. The amplitude of ฮ”ฮฉ is particularly sensitive to ๐›ฟ๐‘Ÿ. km/year for ๐›ฟ๐‘Ÿ = 0.4 (2551 km) in the left plot. The thickening curves in the radial/along-track representation of the relative orbit motion are a further indication of divergence. The strength of divergence in Fig. 4.9 can be directly linked to the size of the offsets in the bounded motion quantities from ๐‘‡๐‘‘โ˜… and ฮ”ฮฉโ˜…, shown in Fig. 4.8. From Fig. 4.8 and Fig. 4.9 we conclude that our method and the resulting expansions in H๐‘ง and ๐ธ for long term bounded motion of at least 10 years around the fixed point LEO from [42] start to lose their significant accuracy for ๐›ฟ๐‘Ÿ โ‰ฅ 0.3 to satisfy the bounded motion conditions with the required precision. Note that ๐›ฟ๐‘Ÿ = 0.3 (1913 km) is already a purely theoretical orbit with altitudes of more than 1000 km below the Earthโ€™s surface, which means that our expansions in H๐‘ง and ๐ธ provided reliable bounded motion beyond realistic distances (๐›ฟ๐‘Ÿ โ‰ค 0.14) between orbits. The behavior of the bounded motion quantities ๐‘‡๐‘‘ and ฮ”ฮฉ for the chosen orbits of the MEO bounded motion set (from Tab. 4.7) are shown in Fig. 4.10. In contrast to the test LEOs, the amplitude and period of oscillation of the bounded motion quantities are decreasing with increasing distance ๐›ฟ๐‘Ÿ, which causes the almost steady behavior of ๐›ฟ๐‘Ÿ = 1.4 over the shown timespan and generally suppresses higher order oscillations that were seen for the LEOs. While the oscillations of 87 6 upper bound of relative distance [1000 km] 5 along-track distance [1000 km] 4 4.5 O0.15 2 O0.20 4 O0.25 O0.30 3.5 0 O0.35 O0.40 3 โˆ’2 2.5 โˆ’4 2 โˆ’6 0 2 4 6 8 10 โˆ’3 โˆ’2 โˆ’1 0 1 2 3 time [years] radial distance [1000 km] Figure 4.9: Distance between the orbits in the calculated bounded motion set and O0 is determined in regular time intervals with numerical integration over more than ten years. The left plot only shows the upper bound to avoid overlaps. Thin horizontal lines at the initial upper bound emphasize small changes. The dotted light blue curve (right) originates from an unintended near-resonance between the chosen time interval for distance evaluations and the orbital behavior. A measurable increase in relative distances (left) over 10 years for ๐›ฟ๐‘Ÿ โ‰ฅ 0.3 is supported by thickening curves in the radial/along-track behavior (right). ๐‘‡๐‘‘ are approximately centered around ๐‘‡๐‘‘โ˜… (except for O1.4 ), the center of oscillation is increasingly diverging from ฮ”ฮฉโ˜… to lower ฮ”ฮฉ for ๐›ฟ๐‘Ÿ โ‰ฅ 0.8. In other words, the expansions in ๐›ฟH๐‘ง and ๐›ฟ๐ธ start failing in producing related orbits that satisfy the bounded motion condition. The consequence of this offset in the bounded motion condition is diverging behavior between the orbits, which can be seen in Fig. 4.11. The upper bound of the total distance between the orbits starts diverging for those very large distances and the thickening curves in the radial/along-track representation of the distance of the orbits from the perspective of O0 further indicate this divergence. Additionally, Fig. 4.11 shows the โ€˜breathingโ€™ in the total relative distance between the orbits with 2๐œ” ๐‘ , which is due to the rotating orbital orientation of the orbits relative to the pseudo-circular fixed point orbit as already mentioned in the section above. From Fig. 4.10 and Fig. 4.11, we conclude that our method and the resulting expansions in H๐‘ง and ๐ธ for long term bounded motion of at least 70 years around the fixed point MEO from [10] 88 53.5398 O0.0 ๐‘‡๐‘‘ [-] 53.5396 O0.6 53.5394 O0.7 O0.8 53.5392 O0.9 O1.0 -0.0192175 O1.1 O1.2 ฮ”ฮฉ [deg] -0.0192180 O1.3 -0.0192185 O1.4 -0.0192190 -0.019220 ฮ”ฮฉ [deg] -0.019225 -0.019230 -0.019235 0 20000 40000 60000 80000 100000 120000 orbital revolutions 0 20 40 60 80 100 120 140 160 approx. time [years] Figure 4.10: Behavior of the bounded motion quantities ๐‘‡๐‘‘ and ฮ”ฮฉ for the test orbits from Tab. 4.7 of the calculated MEO bounded motion set generated by numerical integration. In contrast to the investigated LEOs, the frequency and amplitude of oscillation decrease with increasing ๐›ฟ๐‘Ÿ such that O1.4 appears almost steady. For ๐›ฟ๐‘Ÿ โ‰ฅ 0.8 the center of oscillation of ฮ”ฮฉ start to drift to more negative values and away from ฮ”ฮฉโ˜…. To capture both, the oscillatory behavior around ฮ”ฮฉโ˜… and the drift of the center of oscillation for very large ๐›ฟ๐‘Ÿ, two plots with a different scale and range are shown for ฮ”ฮฉ. start to lose their significant accuracy for ๐›ฟ๐‘Ÿ โ‰ฅ 0.9 to satisfy the bounded motion conditions with the required precision. Interestingly, the very long โ€˜breathingโ€™ periods for very large distances like ๐›ฟ๐‘Ÿ = 1.3 suggested (temporary) bounded motion for the first 70 years when looking at Fig. 4.11, while Fig. 4.10 reveals the underlying diverging behavior due to the mismatched bounded motion conditions. 4.5 Conclusion The normal form methods presented in this chapter yield parameterized sets of the constants of motion ( H๐‘ง (๐›ฟ๐‘Ÿ), ๐ธ (๐›ฟ๐‘Ÿ)) for bounded orbits with an average nodal period and average ascending node drift corresponding to the fixed nodal period and ascending node drift of the reference (fixed 89 2 upper bound of relative distance [10000 km] 1.8 1.5 along-track distance [10000 km] 1.6 1 O0.6 O0.7 0.5 O0.8 1.4 O0.9 O1.0 0 O1.1 1.2 O1.2 โˆ’0.5 O1.3 O1.4 1 โˆ’1 โˆ’1.5 0.8 โˆ’2 0 10 20 30 40 50 60 70 โˆ’1 โˆ’0.5 0 0.5 1 time [years] radial distance [10000 km] Figure 4.11: Distance between the orbits in the calculated bounded motion set and O0 is determined in regular time intervals by numerical integration over more than 70 years. The left plot only shows the upper bound to avoid overlaps. Thin horizontal lines at the initial upper bound emphasize small changes. The โ€˜breathingโ€™ of the total relative distance from the orbital rotation is clearly visible. Its period increases with increasing ๐›ฟ๐‘Ÿ until being unrecognizable due to the strong divergence for ๐›ฟ๐‘Ÿ โ‰ฅ 1.4, which is supported by thinker curves in the right plot. The weaker divergence over the 70-year timespan is already noticeable for ๐›ฟ๐‘Ÿ โ‰ฅ 0.9. The divergence is caused by the offset in respective bounded motion quantities (see Fig. 4.10). point) orbit. The range of ๐›ฟ๐‘Ÿ for which bounded motion orbits can be obtained is dependent on the closeness to the Earth. The closer to the Earth, the stronger the influence of the zonal perturbation on the orbits. Hence, the dynamics of bounded orbits initiated with ๐›ฟ๐‘Ÿ differ much more when they are in a LEO than when they are in a MEO. In comparison to the approach in [42], our method avoided the time-consuming and inaccurate numerical averaging, by using a normal form based parameterization for the averaging. As a result, the range of the bounded motion provided by our methods is more than twice as large as the range of the results in [42]. Additionally, our method does not require a separate calculation for each ๐›ฟ๐‘Ÿ, but rather provides an expansion in (๐›ฟ๐‘Ÿ, ๐›ฟ๐‘ฃ ๐‘Ÿ ), which covers all orbits up to a certain maximum range that varies with the altitude of the reference trajectory. While the method in [10] has the advantage of allowing for the calculation of bounded orbits up 90 to arbitrary distances ๐›ฟ๐‘Ÿ, it lacks the ability to provide parameterized sets of bounded motion just like [42]. The normal form methods are also able to provide parameterized sets of the rotation frequency of the orbits within their orbital plane. This rotation is due to the zonal perturbations in the gravitational field of the Earth since there is no rotation of the orbit for the spherically symmetric case. With increasing distance from the Earthโ€™s center ๐œŒ, the zonal perturbations ๐ฝ๐‘™ fall off with ๐œŒ โˆ’๐‘™โˆ’1 . Accordingly, it is not surprising that the rotation frequency of the MEOs is so much lower than the rotation frequency of the LEOs. Similarly, the ๐›ฟ๐‘Ÿ dependence of the bounded motion is a lot less sensitive for the MEOs compared to the LEOs. 91 CHAPTER 5 STABILITY ANALYSIS OF MUON ๐‘”-2 STORAGE RING This chapter contains parts from my paper Computation and consequences of high order amplitude- and parameter-dependent tune shifts in storage rings for high precision measurements published in the International Journal of Modern Physics A, Vol. 34, No. 36, 1942011 (2019) [96]. The paper was authored by David Tarazona, Martin Berz, and me. The analysis and results from [96] are presented here and they are complemented by additional investigations into period-3 fixed point structures and their relevance in muon loss mechanisms, which was partly discussed in [88]. The differential algebra (DA) map methods (Sec. 2.2) and DA normal form methods (Sec. 2.3) are used to analyze the dynamics of particles in the storage ring of the Muon ๐‘”-2 Experiment at Fermilab. In contrast to the scenarios in Chapter 3 and Chapter 4, this case of study considers two phase space dimensions. We chose a configuration of the storage ring which was utilized during one of the first data-collecting stages. This configuration is particularly interesting because of the closeness to a low-order resonance and its influence on the stability and loss rates of particles. 5.1 Introduction Nonlinear effects of electric and magnetic field components of storage rings to confine the particles and bend their trajectory can cause substantial amplitude dependent tune shifts within the beam. Additionally, tune shifts are often sensitive to variations of system parameters, e.g., offsets of the total particle momentum ๐›ฟ ๐‘ relative to the reference momentum ๐‘ 0 of the storage ring. Such amplitude and parameter dependent tune shifts lead to particles within the beam that oscillate at different frequencies, which potentially affects the beamโ€™s susceptibility to resonances and therefore its dynamics and stability. Thus, it is critical for high precision measurements like the Muon ๐‘”-2 Experiment to analyze and understand these influences. In this chapter, we investigate the dynamics within the Muon ๐‘”-2 Storage Ring, which is the fundamental component of the Muon ๐‘”-2 Experiment, using Poincarรฉ return maps and DA normal 92 form methods. A one-turn Poincarรฉ return map yields the state of particles at a certain azimuthal location within the ring dependent on their state in the previous turn and on system parameters. The application of DA normal form methods to such maps allows for the calculations of the tune shifts and quasi-invariants of the motion around a (stable) fixed point of the map. Additionally, these maps can be used to track the phase space behavior stroboscopically. Before explaining the methods and the results, the following paragraphs will yield a short introduction to the Muon ๐‘”-2 Experiment and its relevance. The goal of the Muon ๐‘”-2 Experiment at Fermilab (E989) [1] is the measurement of the anomalous magnetic dipole moment of the muon ๐‘”๐œ‡ โˆ’ 2 ๐‘Ž๐œ‡ โ‰ก , (5.1) 2 where the ๐‘”-factor relates the spin and magnetic moment of a particle. Dirac theory predicts the factor to be two for charged leptons like the muon [35, 36], but hyperfine structure experiments showed that ๐‘” โ‰  2 [70, 71]. The largest radiative correction was introduced by Schwinger to explain the difference [80, 81]. Over the years more corrections were explored to gain an understanding of the deviation (๐‘”-2, where the name of the experiment comes from) [6]. Today, the most successful theory in particle physics is the standard model (SM). The most accurate calculation of the magnetic dipole moment anomaly of the muon using the standard model, ๐‘Ž SM ๐œ‡ , reaches a precision of 0.39 ppm [6]. The Muon ๐‘”-2 Experiment E821 conducted at the Brookhaven National Laboratory (BNL) yielded a result with a precision of 0.54 ppm [11], which differed from the SM calculation by 3.6 standard deviations. The E989 at Fermilab is the latest experiment in a series of measurements aimed at pushing the precision of the measured result even higher to reach a precision of 0.14 ppm such that the discrepancy between measurement and calculation reaches more than five standard deviations [92]. In this case, the result would be a very strong indication that the standard model is unable to describe this anomaly and would call for adjustments to the model or new theories. The results from the first set of measurements at E989 [1, 4, 2, 3] were in agreement with the measurement from BNL and had a precision of 0.46 ppm. Combined with the result from BNL, this yields an experimental precision of 0.35 ppm and a 93 discrepancy of 4.2 standard deviations from theory predictions [1]. Expectations are that the five standard deviations are reached with the next sets of results from E989. The experimental technique of the Muon ๐‘”-2 Experiment can be briefly summarized as follows [39]: a highly spin-polarized beam of muons is created as a decay product of high energy protons, which decay into pions, which then decay into (positively charged) muons. The muons are delivered through the Muon Campus, which is part of the accelerator complex at Fermilab [87, 83], and injected into the Muon ๐‘”-2 Storage Ring. During the first revolutions within the storage ring, the muon beam is prepared to adjust the emittance and limit fluctuations of the total momentum of the muons to the acceptance range of about ยฑ0.5% relative to the reference momentum ๐‘ 0 . The prepared beam then orbits in the storage ring with only the vertical magnetic field and the four electrostatic quadrupole systems (ESQ) acting on it. The constant magnetic field forces the beam to circle around within the ring and causes the spin of the muons to precess. The four ESQ confine and focus the muons vertically [82]. The muons decay while they orbit and their spins precess. Their decay products, positrons, are measured by the calorimeter system [39] around the beamline in order to determine the spin precession frequency of the muons, which is then used together with a high-precision measurement of the magnetic field [3] to calculate the muon anomalous magnetic moment [11]. Understanding the behavior of the muon beam in the storage ring is particularly important to identify and address problems. One issue is muon loss, which introduces a systematic bias for the average spin precession frequency of the remaining particles, which affects the overall result of the measurement. To better understand the dynamics of the muons and our methods of their analysis, we start with the introduction from [96] into how the Poincarรฉ maps for the storage ring are generated. Then, we discuss the concept of closed orbits and the relevance of the momentum dependent closed orbit. Afterwards, the tune shift analysis from [96] is presented, which serves as the basis of our subsequent investigations into muon loss and the relevance of resonances and their associated fixed point structures. 94 5.2 Storage Ring Simulation Using Poincarรฉ Maps A storage ring is composed of various particle optical elements, each of which can be simulated in COSY INFINITY [61, 26]. For each particle optical element, there is a hypothetical ideal orbit, usually along the center of the element [19]. The ideal orbit is often characterized by a predetermined set of system parameters ๐œ‚ยฎ0 , for example, a specific total reference momentum of the particles. If the element is simulated as ideal, namely without perturbations, the actual trajectory of a particle initiated on the ideal orbit when entering the element (at ๐‘งยฎ0 ) follows the ideal orbit throughout the element. However, with perturbations like imperfections in the associated fields of the element, a particle initiated at ๐‘งยฎ0 might follow a trajectory different from the ideal orbit. Hence, the ideal orbit describes the actual trajectory of a particle initiated at ๐‘งยฎ0 in the unperturbed case. To analyze how an element influences the transverse phase space behavior around the ideal orbit, Poincarรฉ maps (see Sec. 2.2) are used. The Poincarรฉ surfaces correspond to the transverse storage ring cross section perpendicular to the optical axis at azimuthal locations before (S๐‘– ) and after the element (S ๐‘“ ). The Poincarรฉ map P is expanded around the ideal orbit and expresses how the relative phase space state ๐‘งยฎ ๐‘“ โˆˆ S ๐‘“ after the particle optical element depends on variations in the system parameters ๐œ‚ยฎ and on the relative phase space state ๐‘งยฎ๐‘– โˆˆ S๐‘– before the element, with ๐‘งยฎ ๐‘“ = P (ยฎ๐‘ง๐‘– , ๐œ‚). ยฎ The phase space states relative to the ideal orbit ๐‘งยฎ consist of the horizontal (๐‘ž 1 , ๐‘ 1 ) = (๐‘ฅ, ๐‘Ž) and vertical (๐‘ž 2 , ๐‘ 2 ) = (๐‘ฆ, ๐‘) phase space components within the Poincarรฉ surface S. For unperturbed elements, the Poincarรฉ map P is origin preserving, with P ( 0, ยฎ 0) ยฎ = 0, ยฎ since the trajectory follows the ideal orbit. The transverse phase space behavior after a full revolution in the storage ring is given by the Poincarรฉ return map M, which is generated by composing the individual Poincarรฉ maps P๐‘– of the individual storage ring elements according to the storage ring setup (M = P ๐‘˜ โ—ฆ P ๐‘˜โˆ’1 โ—ฆ ... โ—ฆ P2 โ—ฆ P1 ) such that the ideal orbits connect. For the simulation of the Muon ๐‘”-2 Storage Ring, a detailed nonlinear model [85, 86] of the storage ring particle optical elements has been set up using COSY INFINITY. The simulation considers the magnetic field that guides the beam around the storage ring and the four-fold symmetric 95 electrostatic quadrupole system (ESQ) [82], which focuses the beam vertically. The ESQ is not ideal, which makes the simulation of the higher multipole components a critical aspect of the model. Additionally, perturbations due to the ESQ fringe fields and imperfections in the vertical magnetic field can be taken into account based on experimental field measurement data [3]. The model represents the magnetic field inhomogeneities by fitting 2D magnetic multipoles up to fifth order to measurement data of the magnetic field within the Muon ๐‘”-2 Storage Ring (see [3, 85, 86] for details). The ESQ [82] is considered by the corresponding electrostatic potential as a 2D multipole expansion up to tenth order to accurately model the nonlinearities of the system up to the significant contribution of the 20th-pole. The fringe fields of the ESQ โ€“ the fall-off of the electric field at the edges of the ESQ components โ€“ are simulated based on numerical calculations performed with the code COULOMB [91]. The generated Poincarรฉ return maps are expanded in the transverse phase space plane relative to the ideal orbit, where the radial phase space is denoted by (๐‘ฅ, ๐‘Ž) and the vertical phase space is denoted by (๐‘ฆ, ๐‘). The coordinates ๐‘ฅ and ๐‘ฆ indicate the position in the radially outward and vertically upward direction relative to the ideal orbit. The components ๐‘Ž = ๐‘ ๐‘ฅ /๐‘ 0 and ๐‘ = ๐‘ ๐‘ฆ /๐‘ 0 are the momenta perpendicular to the ideal orbit, namely ๐‘ ๐‘ฅ and ๐‘ ๐‘ฆ , scaled by the reference momentum of the particles ๐‘ 0 . Additionally, the maps are expanded in the relative offset ๐›ฟ ๐‘ = ฮ”๐‘/๐‘ 0 with respect to the reference momentum ๐‘ 0 to represent particles with a relative momentum offset. The relative change ๐›ฟ ๐‘ corresponds to the change of the system parameter ๐œ‚. ยฎ To distinguish the effect of various elements of the storage ring and their perturbations on the dynamics of the particles, we simulated different configurations of the components in [96]. Specifically, the influence of perturbations due to ESQ fringe fields and influence from imperfections in the vertical magnetic field were studied separately. We also considered the system for two ESQ voltages, namely 18.3 kV and 20.4 kV. In this chapter of the thesis, however, we will only consider an ESQ voltage of 18.3 kV, since it offers the most interesting nonlinear dynamics and was a set-point used during the first data collection of the Muon ๐‘”-2 Experiment. We are also only considering the map with the magnetic field imperfections since investigations in [96] indicated that it is the 96 dominating perturbation and therefore yields the most realistic results. The main insights from [96] regarding the other cases will still be mentioned at the appropriate places in the text below. 5.3 The Closed Orbit Closed orbits return to themselves after each storage ring revolution, which makes them fixed points of the Poincarรฉ return maps. There are also low period closed orbits that return to themselves after a few turns ๐‘›. These orbits correspond to low period fixed point structures in the ๐‘›-turn Poincarรฉ return map. While there are also unstable fixed points, which are discussed later, we will first focus on the properties of the stables ones. The closed orbit is a reference for the associated particles since they oscillate around it with the closed orbit representing an equilibrium state. Accordingly, the closed orbit is sometimes also referred to as the reference orbit. In the stroboscopic view of the Poincarรฉ return maps, the fixed point mimics an equilibrium point of the oscillatory phase space behavior around it. Using the DA normal form algorithm (see Sec. 2.3) on an origin preserving Poincarรฉ return map, the transverse oscillation frequencies around the fixed point can be calculated. In the rest of this section, we will focus on how these closed orbits and their associated fixed points in the Poincarรฉ return maps are determined. 5.3.1 The Closed Orbit Under Perturbation If all components are simulated to be unperturbed, then the Poincarรฉ return map is a composition of origin preserving Poincarรฉ maps and hence also origin preserving. However, if the simulation considers perturbations, the actual trajectory of the expansion point may be distorted from the ideal orbit and hence not a closed orbit. Accordingly, the expansion point of the associated Poincarรฉ return map may not be a fixed point and the map may not be origin preserving. However, if the perturbation is sufficiently small, a fixed point ๐‘งยฎFP will continue to exist. Parameterizing the strength of the perturbation with ๐œ‚, ยฎ the origin preserving fixed point map of the unperturbed system is given by M (ยฎ๐‘ง, ๐œ‚ยฎ = 0). To analyze the preservation of the param- 97 eter dependent fixed point, an extended map N (ยฎ๐‘ง, ๐œ‚) ยฎ = ( M (ยฎ๐‘ง, ๐œ‚) ยฎ โˆ’ ๐‘งยฎ, ๐œ‚) ยฎ is defined [19]. If det(Jac( N (ยฎ๐‘ง, ๐œ‚)))| ยฎ (ยฎ๐‘ง,๐œ‚)=( ยฎ 0, ยฎ โ‰  0 then, according to the inverse function theorem, an inverse of ยฎ 0) the map N exists for a neighborhood D around the evaluation point ( 0, ยฎ 0) ยฎ of the Jacobian. The parameter dependent fixed point ๐‘งยฎFP ( ๐œ‚)ยฎ of M and hence the closed orbit of the system exists as long as (0, ๐œ‚) ยฎ is within the neighborhood for which invertibility has been asserted. If this is the case and the inverse N โˆ’1 around ( 0,ยฎ 0) ยฎ is given, then the parameter dependent fixed point can be calculated via   (ยฎ๐‘งFP ( ๐œ‚), ยฎ ๐œ‚)ยฎ = N โˆ’1 0, ยฎ ๐œ‚ยฎ . (5.2) Expanding the map around the parameter dependent fixed point yields the origin preserving Poincarรฉ return map under perturbations in the system parameters. The perturbation due to imperfections in the magnetic field distorts particles from the ideal orbit of the E989 storage ring. Accordingly, the Poincarรฉ return map from the composition of the individual particle optical elements is not origin preserving. Using the method above, the fixed point of the map โ€“ the phase space coordinates of the closed orbit at the azimuthal location of the map โ€“ is calculated and the map is expanded around it. The result is an origin preserving fixed point map. Calculating the fixed point for Poincarรฉ return maps at multiple azimuthal locations of the ring indicates the form of the closed orbit (see Fig. 5.1). The collimator locations are highlighted because they are of particular relevance for muon losses. They constitute the narrowest part around the storage region restricting the muons to amplitudes of โˆš๏ธ ๐‘Ÿ = ๐‘ฅ 2 + ๐‘ฆ 2 < 45 mm = ๐‘Ÿ 0 relative to the center of the ring, i.e. the ideal orbit. Muons hitting a collimator during data taking for the measurement are known as lost muons. While the radial/horizontal motion of the closed orbit along the storage ring is close to sinusoidal, the vertical phase space motion is disturbed into more complex behavior. In the ๐‘ฅ๐‘ฆ projection, distorted elliptical motion around the ideal orbit along the center of the ring is indicated. All these deviations from the ideal orbit are triggered by the weak coupling of radial and vertical motion due to ppm-level imperfections of the skew quadrupole magnetic field. The form of the closed orbit is determined by the distribution of such magnetic field imperfections as well as the fields of the ESQ 98 0.6 C5 0.4 C4 C5 0.2 C4 C3 ๐‘ฆ [mm] 0 C3 C2 โˆ’0.2 C1 C1 โˆ’0.4 C2 locations of evaluation โˆ’0.6 within storage ring โˆ’0.6 โˆ’0.4 โˆ’0.2 0 0.2 0.4 0.6 ๐‘ฅ [mm] 0.6 C3 0.6 0.4 0.4 C2 C5 0.2 0.2 C4 C4 C3 ๐‘ฅ [mm] ๐‘ฆ [mm] 0 C1 0 C2 C5 โˆ’0.2 โˆ’0.2 C1 โˆ’0.4 โˆ’0.4 โˆ’0.6 โˆ’0.6 0 60 120 180 240 300 360 0 60 120 180 240 300 360 azimuthal location [deg] azimuthal location [deg] Figure 5.1: The fixed points of Poincarรฉ return maps from various azimuthal locations around the ring indicate the behavior of the closed orbit (for ๐›ฟ ๐‘ = 0). The projections of the four dimensional fixed points into subspaces illustrate the influence of the magnetic field perturbations on the closed orbit around the ring. The results from the five collimator locations (C1-C5) are highlighted with red color. The ๐‘ฅ coordinate corresponds to displacements in the radially outward direction, while the ๐‘ฆ coordinate indicates the displacement in the vertically upward direction. specified by the voltage. The closed orbit we found here and showed in Fig. 5.1 is considering a particle with no momentum offset (๐›ฟ ๐‘ = 0). Following the argumentation above the closed orbit continues to exist with perturbations in ๐›ฟ ๐‘ as will be investigated in the next section. 99 5.3.2 The Momentum Dependence of the Closed Orbit The closed orbit additionally depends on system parameters like the momentum offset of the particles. Just like for the magnetic field perturbation, Eq. (5.2) is used to calculate the parameter dependent fixed point (PDFP) of the origin preserving Poincarรฉ return map, where the parameter is the momentum offset ๐›ฟ ๐‘. The phase space coordinates of the momentum dependent fixed point at the collimator locations in the ring are shown in Fig. 5.2. The primary effect of the momentum offset comes from the interaction of the charged particles 40 C1 0.3 30 C2 C3 0.2 20 C4 ๐‘Ž PDFP [mrad] C5 ๐‘ฅ PDFP [mm] 10 0.1 0 0 โˆ’10 โˆ’0.1 โˆ’20 โˆ’0.2 โˆ’30 โˆ’0.3 โˆ’40 โˆ’0.4 โˆ’0.2 0 0.2 0.4 โˆ’0.4 โˆ’0.2 0 0.2 0.4 ๐›ฟ ๐‘ [%] ๐›ฟ ๐‘ [%] 0.06 0.3 0.05 0.2 ๐‘ PDFP [mrad] 0.04 ๐‘ฆ PDFP [mm] 0.1 0 0.03 โˆ’0.1 0.02 โˆ’0.2 0.01 โˆ’0.3 0 โˆ’0.4 โˆ’0.2 0 0.2 0.4 โˆ’0.4 โˆ’0.2 0 0.2 0.4 ๐›ฟ ๐‘ [%] ๐›ฟ ๐‘ [%] Figure 5.2: Changes of the closed orbits due to relative changes ๐›ฟ ๐‘ in the total initial momentum. The plots illustrate absolute coordinates with respect to the ideal orbit at the center of the ring for the five collimator locations (C1-C5). 100 with the unperturbed part of the vertical magnetic field. The Lorentz force, which determines the orbit radius, is directly proportional to the velocity of the particle, which is relativistically related to the momentum of the particle. This behavior is clearly visible in the horizontal components of Fig. 5.2. The radial position of the parameter dependent fixed point ๐‘ฅ PDFP changes dominantly linearly at about 79 mm/% with the momentum offset at all collimator locations. The associated dependence of the horizontal momentum ๐‘Ž PDFP incorporates the changing radial orientation of the momentum dependent closed orbit with respect to the Poincarรฉ surface and the different orientations at the various collimator locations. The vertical components ๐‘ฆ PDFP and ๐‘ PDFP of the closed orbit are mostly dependent on the azimuthal location of the map and change only slightly with a momentum offset. 5.3.3 The Relevance of Closed Orbits The momentum dependent closed orbits correspond to fixed points in the Poincarรฉ return maps. Particles that are not on a closed orbit oscillate around the momentum dependent closed orbit corresponding to their specific momentum offset. In the stroboscopic view of the Poincarรฉ return maps, this corresponds to stroboscopic oscillatory behavior around the fixed point in both phase spaces in form of distorted ellipses as Fig. 5.3 indicates. Given a particle with distorted elliptical phase space behavior and its corresponding momentum dependent fixed point ๐‘งยฎPDFP (๐›ฟ ๐‘), we define the oscillation amplitudes ๐‘ฅ amp and ๐‘ฆ amp independently from each other. In the radial phase space ๐‘ฅ amp = |๐‘ฅ 0 โˆ’ ๐‘ฅPDFP (๐›ฟ ๐‘)| for ๐‘Ž 0 = ๐‘Ž PDFP (๐›ฟ ๐‘) and ๐‘ฆ amp = |๐‘ฆ 0 โˆ’ ๐‘ฆ PDFP (๐›ฟ ๐‘)| for ๐‘ 0 = ๐‘ PDFP (๐›ฟ ๐‘) in the vertical phase space. The oscillation amplitudes of these transverse oscillations are determined by the phase space position of the particle and the momentum dependent fixed point. Particles with the same oscillation amplitudes but different momentum offsets will follow roughly the same motion, but at different locations in phase space. On the other hand, particles at the same phase space location may follow entirely different orbital motion depending on their corresponding momentum dependent fixed point. In summary, the phase space motion of a particle is characterized by its momentum dependent fixed 101 6 40 40 4 30 30 20 20 2 ๐‘Ž [mrad] 10 10 y [mm] y [mm] 0 0 0 โˆ’10 โˆ’10 โˆ’2 โˆ’20 โˆ’20 โˆ’4 P1 P3 โˆ’30 โˆ’30 P2 P4 โˆ’40 โˆ’40 โˆ’6 โˆ’40 โˆ’20 0 20 40 โˆ’1 0 1 โˆ’40 โˆ’20 0 20 40 ๐‘ฅ [mm] b [mrad] x [mm] Figure 5.3: Phase space behavior of four particles in different phase space regions with various amplitudes and momentum offsets. Particle 4 (yellow) hits the collimator (circle in the ๐‘ฅ๐‘ฆ plot) and is lost. The momentum dependent radial position ๐‘ฅ of the particles is particularly  prominent. The individual particles are characterized by the parameter set ๐‘ฅ amp , ๐‘ฆ amp , ๐›ฟ ๐‘ . For particle 1 (P1) the parameter set is (6 mm, 12 mm, โˆ’0.39%). For particle 2 (P2) the parameter set is (12 mm, 6 mm, โˆ’0.39%) . For particle 3 (P3) the parameter set is (27 mm, 16 mm, +0.13%). For particle 4 (P4) the parameter set is (6 mm, 25 mm, +0.39%). point, its amplitudes of oscillation, and its oscillation frequencies, which are addressed in detail in Sec. 5.4. The collimators restrict the maximum amplitudes of oscillation around the associated momentum dependent fixed points. Fig. 5.4 illustrates the shape of the viable phase space region around a momentum dependent fixed point. The closeness of the reference closed orbit to the collimators increases the risk of muon loss. While particles with low momentum offset are only at risk of getting lost when they have relatively large oscillation amplitudes, particles with a large momentum offset may already be lost with seemingly small amplitudes of oscillation. Since the semi-major and semi-minor axis of the distorted elliptical phase space behavior are not necessarily aligned with the position and momentum axis and vary for each particle, there is no straightforward definition of the amplitude of oscillation. The DA normal form algorithm takes care of this by transforming the distorted ellipses in phase space to circles such that the amplitudes of oscillation are just the radii of the circles โ€“ the normal form radii. We will investigate the relationship between the original phase space coordinates and the normal form radii more closely later on and also use its advantages, but for now, we want to focus on practically relatable quantities in the original phase space, rather than abstract quantities like the normal form radius. 102 Figure 5.4: Schematic illustration of viable ๐‘ฅ๐‘ฆ region around a momentum dependent fixed point. The region contains all rectangles centered at the fixed point, which do not overlap with the collimator circle. 5.4 Tune Analysis The following tune analysis investigates the oscillation frequency around the reference closed orbits depending on the momentum offset and the amplitude of oscillation. The tunes shall shed light on average loss times and the involvement of resonances. 5.4.1 Tunes of the Momentum Dependent Closed Orbit Given the parameter dependent fixed point map representing the phase space behavior around the momentum dependent closed orbit of the Muon ๐‘”-2 Storage Ring model, the diagonalization in the DA normal form algorithm is used to determine the tunes of the momentum dependent closed orbit. The calculated tunes of the closed orbit (for ๐›ฟ ๐‘ = 0) differ only very slightly depending on the azimuthal location of the Poincarรฉ return map yielding ๐œˆ๐‘ฅ = 0.944462633(8 ยฑ 3) and ๐œˆ ๐‘ฆ = 0.330814444(7 ยฑ 6), (5.3) which is expected since they all describe the linear motion around same closed orbit. The proximity of the vertical tune ๐œˆ ๐‘ฆ to the low 1/3-resonance will be investigated more closely later. The radial 103 tune ๐œˆ๐‘ฅ is even closer to a higher order resonance namely the 17/18-resonance. Without loss of generality, we will use the Poincarรฉ return map at collimator C3 for our further map investigations. The Fig. 5.5 illustrates the momentum dependence of the tunes over the momentum offset range of ๐›ฟ ๐‘ โˆˆ [โˆ’0.5%, 0.5%] and indicates the linear dependence (chromaticities) ๐œ‰๐‘– as a reference. For |๐›ฟ ๐‘| < 0.25% the momentum dependence of both tunes is predominantly linear with ๐œ‰๐‘ฅ = โˆ’0.131999346 and ๐œ‰ ๐‘ฆ = 0.389753993. (5.4) For |๐›ฟ ๐‘| > 0.33% however, the tunes are dominated by an order eight dependence on relative momentum offsets ๐›ฟ ๐‘. This eighth order dependence results from the strong ninth order terms in the original map, which are linear in the phase space components and of order eight in the momentum dependence, representing the earlier mentioned significant influence of the 20th-pole of the ESQ potential (see Sec. 5.2). Interestingly, the linear coefficient and the eighth order coefficient of the vertical momentum dependent tune shifts are both larger by a factor of three and opposite in sign compared to their radial counterparts. Additionally, the momentum dependent vertical tune shifts away from the 1/3-resonance. The investigation in [96] indicated a strong influence of the ESQ voltage on the linear motion 0.962 0.335 0.96 ๐œˆ๐‘ฅ = ๐œ– ๐‘ฅ ๐›ฟ ๐‘ 0.33 0.958 ๐œˆ๐‘ฅ (๐›ฟ ๐‘) 0.325 0.956 0.32 0.954 0.315 0.31 ๐œˆ๐‘ฅ 0.952 ๐œˆ๐‘ฆ 0.305 0.95 0.3 0.948 0.295 0.946 0.29 ๐œˆ๐‘ฆ = ๐œ– ๐‘ฆ ๐›ฟ ๐‘ 0.944 0.285 ๐œˆ ๐‘ฆ (๐›ฟ ๐‘) 0.942 0.28 โˆ’0.4 โˆ’0.2 0 0.2 0.4 โˆ’0.4 โˆ’0.2 0 0.2 0.4 ๐›ฟ ๐‘ [%] ๐›ฟ ๐‘ [%] Figure 5.5: Vertical and horizontal tune dependence in the model of the Muon ๐‘”-2 Storage Ring of E989 on relative offsets ๐›ฟ ๐‘ from the reference momentum ๐‘ 0 . 104 around the respective expansion points and therefore the tunes. The momentum dependence of the tunes โ€“ the momentum dependent tune shifts โ€“ however is only slightly changed by the ESQ voltage (see [96] for more details). 5.4.2 The Amplitude Dependent Tune Shifts The DA normal form algorithm provides the transformation ANF from the original phase space coordinates (๐‘ฅ, ๐‘Ž) and (๐‘ฆ, ๐‘) to rotationally invariant normal form coordinates (๐‘ž NF,1 , ๐‘ NF,1 ) and (๐‘ž NF,2 , ๐‘ NF,2 ). The amplitude and parameter dependent tune shifts ๐œˆ๐‘– (๐‘Ÿ NF,1 , ๐‘Ÿ NF,2 , ๐›ฟ ๐‘) can be extracted from the normal form map, where the amplitudes are given by the normal form radii โˆš๏ธƒ ๐‘Ÿ NF,๐‘– = ๐‘ž 2NF,๐‘– + ๐‘ 2NF,๐‘– . This full description of the tunes and their dependence on phase space amplitudes and momentum offsets is extremely powerful. However, the abstract normal form radii are not as practically useful as the previously defined oscillation amplitudes ๐‘ฅ amp and ๐‘ฆ amp in original phase space coordinates. To address this, Fig. 5.6 illustrates the dependence of the tunes on the radial phase space amplitude ๐‘ฅ amp and the dependence on the vertical phase space amplitude ๐‘ฆ amp , separately. This is done by calculating the corresponding normal form coordinates and normal form radii and using those for the tune evaluation. The amplitude dependence is never linear but always appears as even orders. Investigations in [96] indicated that amplitude dependent tune shifts, just like momentum dependent tune shifts, are only weakly influenced by the ESQ voltages and the field perturbations. Similar to the purely momentum dependent tune shifts, the sign of the momentum offset seems to only play a minor role compared to the magnitude of the offset. The radial amplitude dependence of the tunes is relatively well behaved. Again, there is the dominating eighth order dependence related to the strong ninth order nonlinear terms resulting from the 20th-pole of the ESQ potential, which shifts the tunes of the radial phase space up and tunes of the vertical phase space down with increasing radial amplitude and magnitude of the momentum offset. 105 0.96 0.96 ๐›ฟ ๐‘ [%] 0.958 0.5 0.955 0.956 0.954 0.95 0.4 ๐œˆ๐‘ฅ 0.952 ๐œˆ๐‘ฅ 0.945 0.95 0.3 0.94 0.948 0.935 0.2 0.946 0.944 0.93 0.1 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 ๐‘ฅ amp [mm] ๐‘ฆ amp [mm] 0 0.33 โˆ’0.1 0.34 0.32 0.33 โˆ’0.2 0.31 0.32 โˆ’0.3 ๐œˆ๐‘ฆ ๐œˆ๐‘ฆ 0.3 0.31 โˆ’0.4 0.29 0.3 โˆ’0.5 0.28 0.29 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 ๐‘ฅ amp [mm] ๐‘ฆ amp [mm] Figure 5.6: Amplitude dependent tune shifts in the model of the Muon ๐‘”-2 Storage Ring of E989. The black line indicates the amplitude dependent tune shifts for ๐›ฟ ๐‘ = 0, while the other lines have a momentum offset specified by their color. For the left plots regarding the radial amplitude dependence, the vertical amplitude relative to the momentum dependent fixed point is set to zero and vice versa for the plots regarding the vertical amplitude dependence on the right. The lines end when the total ๐‘ฅ๐‘ฆ amplitude of the particle relative to the ideal orbit reaches the collimator at ๐‘Ÿ 0 = 45 mm. The vertical amplitude dependence however is more complex as it varies strongly with the magnitude of the momentum offset. Regarding the vertical tune, this is particularly critical due to the crossing of the 1/3-resonance tune for some vertical amplitude and momentum offset combinations. Such low resonances can have a major influence on the dynamics of particles which is why we will closely investigate these cases later. Even though the purely momentum dependent tune shifts (๐‘ฅamp = 0, ๐‘ฆ amp = 0) and the tune 106 shifts purely dependent on the vertical amplitude (๐‘ฅ amp = 0, ๐›ฟ ๐‘ = 0) shift in the same direction โ€“ up for radial tunes and down for vertical tunes โ€“ there are opposing cross-terms, which depend both on the vertical amplitude and the momentum offset that trigger this nontrivial tune shift behavior. In Fig. 5.7 to Fig. 5.9 the combined effects of simultaneous radial and vertical amplitudes on the tune shifts are illustrated for selected momentum offsets. The behavior for the intermediate momentum offsets may be interpolated from the given plots. Again, the sign of the momentum offset has only a minor influence on the form of the tune shifts compared to its magnitude. Note that Fig. 5.7 to Fig. 5.9 only illustrates tunes for phase space states within the viable phase space around the corresponding momentum dependent fixed point. Accordingly, not all lines extend over the full 45 mm range of ๐‘ฆ amp and some lines for large ๐‘ฅamp are not shown, since their total ๐‘ฅ๐‘ฆ amplitude of the particle relative to the ideal orbit reaches the collimator at ๐‘Ÿ 0 = 45 mm. The combined effects in Fig. 5.7 to Fig. 5.9 emphasize the strong nonlinear character of the tune dependencies, which was already indicated in Fig. 5.6. The wave-like structure illustrates how different order terms dominate at different vertical amplitudes ๐‘ฆ amp depending on both, the radial amplitude ๐‘ฅ amp and the momentum offset ๐›ฟ ๐‘. Additionally, for almost every momentum offset there are combinations of oscillation amplitudes for which the vertical 1/3-resonance tune is crossed. Investigations in [96] did not show this strong nonlinear behavior of the combined effects on the tune shifts in such clarity. 107 ๐‘ฅamp 0.96 ๐›ฟ ๐‘ = 0.40% 0.335 0.955 0.33 0.325 45 0.95 ๐œˆ๐‘ฅ ๐œˆ๐‘ฆ 0.32 0.945 0.315 0.94 40 0.31 0.935 0.305 ๐›ฟ ๐‘ = 0.40% 0.96 ๐›ฟ ๐‘ = 0.32% 0.335 35 0.955 0.33 0.325 0.95 30 ๐œˆ๐‘ฅ ๐œˆ๐‘ฆ 0.32 0.945 0.315 0.94 0.31 25 0.935 0.305 ๐›ฟ ๐‘ = 0.32% 0.96 ๐›ฟ ๐‘ = 0.26% 0.335 20 0.955 0.33 0.325 0.95 ๐œˆ๐‘ฅ ๐œˆ๐‘ฆ 0.32 15 0.945 0.315 0.94 0.31 10 0.935 0.305 ๐›ฟ ๐‘ = 0.26% 0.96 ๐›ฟ ๐‘ = 0.14% 0.335 5 0.955 0.33 0.325 0.95 ๐œˆ๐‘ฅ ๐œˆ๐‘ฆ 0.32 0.945 0 0.315 0.94 0.31 0.935 0.305 ๐›ฟ ๐‘ = 0.14% 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 ๐‘ฆ amp [mm] ๐‘ฆ amp [mm] Figure 5.7: Behavior of combined amplitude dependent tune shifts at multiple momentum offsets for an ESQ voltage of 18.3 kV. 108 ๐‘ฅamp 0.96 ๐›ฟ ๐‘ = 0.40% 0.335 0.955 0.33 0.325 45 0.95 ๐œˆ๐‘ฅ ๐œˆ๐‘ฆ 0.32 0.945 0.315 0.94 40 0.31 0.935 0.305 ๐›ฟ ๐‘ = 0.40% 0.96 ๐›ฟ ๐‘ = 0.32% 0.335 35 0.955 0.33 0.325 0.95 30 ๐œˆ๐‘ฅ ๐œˆ๐‘ฆ 0.32 0.945 0.315 0.94 0.31 25 0.935 0.305 ๐›ฟ ๐‘ = 0.32% 0.96 ๐›ฟ ๐‘ = 0.26% 0.335 20 0.955 0.33 0.325 0.95 ๐œˆ๐‘ฅ ๐œˆ๐‘ฆ 0.32 15 0.945 0.315 0.94 0.31 10 0.935 0.305 ๐›ฟ ๐‘ = 0.26% 0.96 ๐›ฟ ๐‘ = 0.14% 0.335 5 0.955 0.33 0.325 0.95 ๐œˆ๐‘ฅ ๐œˆ๐‘ฆ 0.32 0.945 0 0.315 0.94 0.31 0.935 0.305 ๐›ฟ ๐‘ = 0.14% 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 ๐‘ฆ amp [mm] ๐‘ฆ amp [mm] Figure 5.8: Behavior of combined amplitude dependent tune shifts at multiple momentum offsets for an ESQ voltage of 18.3 kV. 109 ๐‘ฅamp 0.96 ๐›ฟ ๐‘ = 0.40% 0.335 0.955 0.33 0.325 45 0.95 ๐œˆ๐‘ฅ ๐œˆ๐‘ฆ 0.32 0.945 0.315 0.94 40 0.31 0.935 0.305 ๐›ฟ ๐‘ = 0.40% 0.96 ๐›ฟ ๐‘ = 0.32% 0.335 35 0.955 0.33 0.325 0.95 30 ๐œˆ๐‘ฅ ๐œˆ๐‘ฆ 0.32 0.945 0.315 0.94 0.31 25 0.935 0.305 ๐›ฟ ๐‘ = 0.32% 0.96 ๐›ฟ ๐‘ = 0.26% 0.335 20 0.955 0.33 0.325 0.95 ๐œˆ๐‘ฅ ๐œˆ๐‘ฆ 0.32 15 0.945 0.315 0.94 0.31 10 0.935 0.305 ๐›ฟ ๐‘ = 0.26% 0.96 ๐›ฟ ๐‘ = 0.14% 0.335 5 0.955 0.33 0.325 0.95 ๐œˆ๐‘ฅ ๐œˆ๐‘ฆ 0.32 0.945 0 0.315 0.94 0.31 0.935 0.305 ๐›ฟ ๐‘ = 0.14% 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 ๐‘ฆ amp [mm] ๐‘ฆ amp [mm] Figure 5.9: Behavior of combined amplitude dependent tune shifts at multiple momentum offsets for an ESQ voltage of 18.3 kV. 110 5.4.3 The Tune Footprint The tune footprint visualizes the projection of a beam distribution into tune space. The COSY INFINITY based model [85, 86] of the Muon ๐‘”-2 Storage Ring is used to generate the realistic beam distribution of 37738 particles from orbit tracking of the muon beam until it is circulating in the storage ring, prepared for data analysis. In particular, the model considers the imperfect injection process, which attempts to align the injected beam with the ideal orbit of the storage ring as well as possible. The model also considers the mispowered ESQ components to imitate the preparation mechanism during the first turns of the beam in the storage ring at E989. Further details of the tracking model and on how a realistic distribution of particles is obtained are elaborated in [85, 86]. The variables (๐‘ฅ, ๐‘Ž, ๐‘ฆ, ๐‘, ๐›ฟ ๐‘) relative to the ideal orbit are illustrated in Fig. 5.10 as projections into the (๐‘ฅ, ๐‘Ž), (๐‘ฆ, ๐‘), and (๐‘ฅ, ๐‘ฆ) subspaces. The beam distribution tends towards higher total momenta in the range of ๐›ฟ ๐‘ โˆˆ [โˆ’0.2%, 0.4%] while overall staying well within the momentum acceptance range of ยฑ0.5%. The spread of the vertical momentum component ๐‘ is about a factor two to three smaller than its horizontal counterpart ๐‘Ž. The position space (๐‘ฅ๐‘ฆ) is filled up to the limitations due to the collimators. The distributions of the horizontal and vertical tunes are illustrated by the tune footprint in Fig. 5.11, where the vertical tunes of the particle distribution are plotted against their horizontal tunes as previously done in [56]. The tune footprint of the tenth order calculation is overlaid by the result of an eighth order calculation to emphasize the influence of the strong ninth order nonlinearities of the map caused by the 20th-pole of the ESQ potential. The tune footprint of the tenth order calculation is five to six times larger in each dimension than its eighth order counterpart. Additionally, particles in different momentum offset ranges are highlighted to illustrate the behavior of this specific group. The tune footprint can be segmented into three groups characterized by their momentum offset which generates a tune footprint in the shape of a โ€˜Tโ€™. The tune footprint for the other ESQ voltages in [96] has a similar distribution for the order eight and order ten calculations, respectively. While the reference tunes are mainly determined by the ESQ voltage, the relative tune shifts behave very similarly. If the ESQ voltage were to place the 111 40 40 ๐›ฟ ๐‘ [%] 30 30 20 20 0.4 10 10 ๐‘ฆ [mm] ๐‘ฆ [mm] 0 0 -10 -10 0.3 -20 -20 -30 -30 0.2 -40 -40 -40-30-20-10 0 10 20 30 40 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 ๐‘ฅ [mm] ๐‘ [mrad] 0.1 6 12 0 4 10 2 8 Ratio [%] -0.1 ๐‘Ž [mrad] 0 6 -2 4 -0.2 -4 2 -6 0 -40-30-20-10 0 10 20 30 40 -0.4 -0.2 0 0.2 0.4 ๐‘ฅ [mm] ๐›ฟ ๐‘ [%] Figure 5.10: Projections of the distribution of the variables (๐‘ฅ, ๐‘Ž, ๐‘ฆ, ๐‘, ๐›ฟ ๐‘) in the realistic beam simulation at the azimuthal ring location of the central kicker. reference tunes very close to a resonance line, we expect the tune distribution and tune shifts to behave differently. Fig. 5.11 shows that the vertical 1/3-resonance tune cannot only be reached hypothetically for the apparent case of a nominal set-point away from resonances. A substantial part of particles is close to or on this low order resonance. The overlaid eighth order calculation shows that this is triggered by the strong ninth order nonlinearities of the map caused by the 20th-pole of the ESQ potential. The segmentation with regard to the momentum offset of the particles into subgroups additionally shows that the vertical 1/3-resonance tune is crossed in each of those groups. The resonance point (17/18, 1/3) is also covered and surrounded by many particles and might have a 112 a) b) 0.335 0.335 ๐›ฟ ๐‘ [%] 0.33 0.33 vertical tunes ๐œˆ ๐‘ฆ vertical tunes ๐œˆ ๐‘ฆ 0.4 0.325 0.325 0.3 0.32 0.32 0.315 0.315 0.2 0.31 0.31 0.94 0.945 0.95 0.94 0.945 0.95 0.1 radial tunes ๐œˆ๐‘ฅ radial tunes ๐œˆ๐‘ฅ c) d) 0.335 0.335 0 0.33 0.33 vertical tunes ๐œˆ ๐‘ฆ vertical tunes ๐œˆ ๐‘ฆ -0.1 0.325 0.325 0.32 0.32 -0.2 0.315 0.315 0.31 0.31 0.94 0.945 0.95 0.94 0.945 0.95 radial tunes ๐œˆ๐‘ฅ radial tunes ๐œˆ๐‘ฅ Figure 5.11: The tune footprint of a realistic beam distribution at the azimuthal ring location of the central kicker. The tune footprint from the 10th order calculation is colored according to the momentum offset of the individual particles. The black lines correspond to resonance conditions. In a) the 8th order calculation (green) is overlaid to illustrate the drastic influence of the strong ninth order nonlinearities of the map caused by the 20th-pole of the ESQ potential. In b) the particles with a momentum offset โˆ’0.3% < ๐›ฟ ๐‘ < 0.1% are overlaid in green. In c) the particles with a momentum offset 0.1% < ๐›ฟ ๐‘ < 0.28% are overlaid in green. In d) the particles with a momentum offset 0.28% < ๐›ฟ ๐‘ < 0.5% are overlaid in green. particularly strong impact. 5.5 Stability and Loss Mechanisms Muons are lost when they hit structural parts of the storage ring and lose the energy necessary to remain within the storage region during data taking. Collimators, which are inserted at various 113 azimuthal locations in the ring (see Fig. 5.1), constitute the narrowest part around the storage region. โˆš๏ธ They restrict the muons to amplitudes of ๐‘Ÿ = ๐‘ฅ 2 + ๐‘ฆ 2 < 45 mm = ๐‘Ÿ 0 relative to the ideal orbit. Our previous analysis is very helpful for gaining a general understanding of certain properties of the system, e.g. the momentum dependence of the reference orbit, and the momentum and amplitude dependent shifts in the oscillation frequency of orbits around their repetitive reference orbit. This analysis showed that the vertical 1/3-resonance tune is relevant for various combinations of amplitudes and momentum offsets. However, only tracking analysis can yield the actual phase space behavior of lost particles and particles involved with the vertical 1/3-resonance tune. Additionally, we saw that the radial tune is very close to the high order 17/18-resonance, which we will also look at more closely. For the tracking analysis, we use both one-turn maps as well as sectional maps. The one-turn Poincarรฉ return map yields the state of a muon at the azimuthal location of the central kicker depending on its state in the previous turn. Sectional maps transfer the state of a muon to the azimuthal location of the collimators. Accordingly, the muons are not tracked continuously, but stroboscopically at specific azimuthal locations e.g. at the respective collimator locations. There are two common approaches for tracking analysis. For a general understanding of the phase space dynamics of the storage ring, one could track a particle distribution, which is evenly distributed in all phase space dimensions and over momentum offset range. However, the implication from such an analysis for the actual muon beam might be limited, since the actual muon beam is not evenly distributed. Accordingly, we track the realistic particle distribution of 37738 particles from Sec. 5.4.3. We take the distribution of particles at the central kicker after 200 turns (corresponding to about 30 ๐œ‡s) from the injection, which is when data taking begins. During this initial 30 ๐œ‡s after injection, the muon beam and the system are conditioned for data taking [2]. The beam is tracked for additional 4500 turns (670 ๐œ‡s), while determining and recording various orbit parameters that shall be analyzed in detail below. 114 5.5.1 The Normal Form Defect of Tracked Particles As explained in Sec. 2.4, the normal form defect yields the inaccuracies in the normal form, i.e., how much the pseudo-invariants (the normal form radii) vary per turn. Using tracking simulations, one can evaluate a related quantity that we will call the long term normal form defect ๐‘‘NF,lt = ๐‘Ÿ NF,max โˆ’ ๐‘Ÿ NF,min . (5.5) It yields the difference between the maximum and the minimum normal form radius of a single particle orbit over the many turns of the long term tracking. Thus, provides the normal form radius range of the orbital pattern. The maximum per turn normal form defect ๐‘‘NF,max of the particle is the maximum rate of change of the normal form radius during the orbital pattern. In Fig. 5.12 the particles are grouped by the maximum per turn normal form defect they encountered during the 4500 turns of tracking. The rate of particles getting lost is strongly correlated with the size of the maximum normal form defect they encountered. More than 91% of particles 0.1 1 512 ratio of particles with larger ๐‘‘NF,max 0.9 256 0.8 128 0.01 0.7 64 ๐‘‘NF,lt /๐‘‘NF,max 0.6 loss ratio particle ratio 32 loss ratio 0.5 16 0.4 0.001 8 0.3 0.2 4 0.1 2 0.0001 0 1 โˆ’11 โˆ’10 โˆ’9 โˆ’8 โˆ’7 โˆ’6 โˆ’5 โˆ’4 survive lost ๐‘‘NF,max [log2 ] Figure 5.12: The left plot shows in violet the ratio of particles that have a larger ๐‘‘NF,max then the corresponding ๐‘ฅ value. The green boxes indicate the ratio of particles lost in each ๐‘‘NF,max group, e.g., particles that encounter a maximum normal form defect larger than 2โˆ’7 = 7.8 ยท 10โˆ’3 are all lost (loss ratio of 1). The right plot shows the ratio between the normal form radius range over 4500 turns (๐‘‘NF,lt ) and the maximum encountered normal form defect ๐‘‘NF,max for each particle. The particles are grouped into surviving and lost particles. 115 have a maximum normal form defect smaller than 2โˆ’11 = 4.9 ยท 10โˆ’4 and none of those particles is lost. With larger maximum normal form defects, the loss rate increases significantly. Particles that encounter a maximum normal form defect larger than 2โˆ’7 = 7.8 ยท 10โˆ’3 are all lost. This confirms that the size of the per turn normal form defect is a good indication for losses, in the sense that the larger the per turn normal form defect the more likely the particle gets lost. The right side of Fig. 5.12 illustrates the ratio of the long term normal form defect ๐‘‘NF,lt to the maximum per turn normal form defect ๐‘‘NF,max of a particle. Considering that ๐‘‘NF,lt over 4500 turns is only a factor of eight to 16 larger than ๐‘‘NF,max for surviving particles illustrates the overestimation of Nekhoroshev-type stability estimates (see Sec. 2.4) based on the per turn normal form defect. Additionally, the ratio is much more shifted to higher factors for lost particles, indicating less overestimation for lost particles with Nekhoroshev-type stability estimates. In Fig. 5.13 the relevance of the resonances โ€“ especially low order resonances like the vertical 1/3-resonance tune โ€“ on the long term normal form defect becomes obvious. Since the tunes are dependent on the normal form radii, a larger long term normal form defect automatically corresponds to a larger tune range of a particle. 0.04 0.3 tune range tune range 0.035 ๐œˆ๐‘ฅ,min (all) 0.25 ๐‘ฆ,min (all) ๐œˆ 0.03 ๐œˆ ๐‘ฅ,min (lost) ๐œˆ ๐‘ฆ,min (lost) 0.025 0.2 ๐‘‘NF,lt,๐‘ฅ 0.02 ๐‘‘NF,lt,๐‘ฆ 0.15 0.015 0.1 0.01 0.005 0.05 0 0 0.93 0.94 0.95 0.96 0.32 0.33 0.34 radial tune ๐œˆ๐‘ฅ vertical tune ๐œˆ ๐‘ฆ Figure 5.13: The plots show the long term normal form defect dependent on the calculated tune range of each particle. The dots are the minimum calculated tune of each particle while tracking. Red dots indicate that the respective particle is lost over the 4500 tracking turns. The gray lines show the calculated tune range of each particle. The left plot illustrates the radial long term normal form defect with respect to the radial tune and the 17/18 resonance (green line). The right plot shows the vertical long normal form defect with respect to the vertical tune and the 1/3 resonance (green line). 116 In the plot of the vertical tune against the vertical long term normal form defect, there is a โ€˜spikeโ€™ facing roughly 45โ—ฆ away from the resonance line. In Fig. 5.14, the tune range of these โ€˜spikeโ€™ particles is analyzed to determine a resonance as a potential trigger of the increasing normal form defect. The analysis indicates that the 10th order 6๐œˆ๐‘ฅ + 4๐œˆ ๐‘ฆ = 7 resonance might be the cause of this spike, but it remains unknown why the normal form defect increases along this resonance with increasing distance from the 1/3-resonance. The normal form radii are the oscillation amplitudes in the high order normalized, linearly decoupled phase space. They are closely related to the oscillation amplitudes in the respective phase spaces relative to the momentum dependent closed orbit. The strong variation in the normal form radii (the large long term normal form defect) of some orbits indicates that the corresponding oscillation amplitude of those orbits around their respective reference orbits is also not constant. To investigate this more closely, the following section investigates the orbits of all lost particles. 0.07 0.331 0.06 0.05 0.33 ๐‘‘NF,lt,๐‘ฆ 0.04 ๐œˆ๐‘ฆ 0.329 0.03 0.02 0.328 0.01 6๐œˆ๐‘ฅ + 4๐œˆ ๐‘ฆ = 7 0.327 0 0.942 0.945 0.948 0.951 โˆ’0.02 0 0.02 ๐œˆ๐‘ฅ 6๐œˆ๐‘ฅ + 4๐œˆ ๐‘ฆ โˆ’ 7 Figure 5.14: The tune range of the particles forming the spike in Fig. 5.13 are shown on the left. The right plot shows the normal form defect of the particles depends on their closeness to the 6๐œˆ๐‘ฅ + 4๐œˆ ๐‘ฆ = 7 resonance (green line). 5.5.2 Lost Muon Studies In this section, we track and investigate all 259 muons of the distribution from Sec. 5.4.3 that are lost at collimator C3 and/or C4 over the 4500 turns. In Fig. 5.15 to Fig. 5.29, 15 of those lost particles are picked to illustrate the different phase space behaviors observed for lost particles. Each figure illustrates the behavior of a different particle and is made up of six plots. The scaling of the 117 plots is the same for all figures. In each figure, the left two plots show the radial and vertical phase space behavior. The ๐‘ฅ๐‘ฆ behavior is shown in the top center plot above the normal form phase space โˆš๏ธƒ behavior (๐‘ž NF,2 , ๐‘ NF,2 ). The top right plot shows the normal form radius ๐‘Ÿ NF = ๐‘Ÿ NF,1 2 2 + ๐‘Ÿ NF,2 over the number of turns, and the bottom right plot shows the tune footprint of the particle. In the caption, the momentum offset of the particle is mentioned. One striking property that many of the lost muons share is the appearance of threefold-symmetry patterns in the vertical phase space projections. The calculated tunes of these lost particles are all crossing or proceed very close to the vertical 1/3-resonance. These threefold-symmetry patterns often include significant modulations in the vertical oscillation amplitude, which is additionally shown by the changing overall normal form radius ๐‘Ÿ NF and the variations in the calculated tunes shown in the tune footprint. While there are many patterns, there are two that stick out, namely, the island pattern (see for example Fig. 5.17) and the shuriken pattern (see for example Fig. 5.23). In Sec. 5.5.3, we will understand how all these patterns are related to period-3 fixed point structures. The patterns come in stable, semi-stable, and unstable forms. This tendency to unstable behavior is often associated with a large radial amplitude and/or a closeness to the (๐œˆ๐‘ฅ , ๐œˆ ๐‘ฆ ) = (17/18, 1/3) resonance point. The โ€˜fuzzynessโ€™ of the vertical phase space pattern in (๐‘ฆ, ๐‘) compared to the pattern in the corresponding normal form phase space (๐‘ž NF,2 , ๐‘ NF,2 ) is related to the radial phase space motion. Due to the weak coupling between the radial and vertical phase space from the imperfections in the magnetic field, large amplitudes in (๐‘ฅ, ๐‘Ž) notably affect the motion in (๐‘ฆ, ๐‘), which does not happen in the decoupled normal form phase space. This โ€˜fuzzynessโ€™ might also trigger the jumping between different patterns for orbits, which are close to the border between two patterns (see Fig. 5.31). While studying the figures below, pay attention to the modulation frequency of the vertical amplitude / the normal form radius. Shuriken and unstable patterns yield the slowest modulations followed by large and small island patterns. The fastest modulations occur in almost regularly looking patterns in form of distorted ellipses. There also seems to be a correlation between the size of the modulation and its frequency, i.e., the larger the modulation the slower its frequency. 118 1.1 6 40 1 normal form radius ๐‘Ÿ NF 4 20 0.9 2 ๐‘Ž [mrad] ๐‘ฆ [mm] 0.8 0 0 0.7 -2 -20 0.6 -4 0.5 -40 -6 0.4 -40 -20 0 20 40 -40 -20 0 20 40 1000 2000 3000 4000 ๐‘ฅ [mm] ๐‘ฅ [mm] turns 0.34 1 2 0.338 1 0.5 0.336 vertical tune ๐œˆ ๐‘ฆ ๐‘ [mrad] 0.334 0 ๐‘ NF,2 0 0.332 -1 -0.5 0.33 0.328 -2 -1 0.326 -40 -20 0 20 40 -1 -0.5 0 0.5 1 0.935 0.94 0.945 0.95 ๐‘ฆ [mm] ๐‘ž NF,2 radial tune ๐œˆ๐‘ฅ Figure 5.15: The radial and vertical phase space behavior indicates that this particle (๐›ฟ ๐‘ = 0.015%) oscillates at constant amplitudes around its momentum dependent reference orbit. The overall normal form radius is constant and confirms this. Accordingly, the tune footprint of the particle is a single dot. This is a trivial large amplitude loss. 119 1.1 6 40 1 normal form radius ๐‘Ÿ NF 4 20 0.9 2 ๐‘Ž [mrad] ๐‘ฆ [mm] 0.8 0 0 0.7 -2 -20 0.6 -4 0.5 -40 -6 0.4 -40 -20 0 20 40 -40 -20 0 20 40 1000 2000 3000 4000 ๐‘ฅ [mm] ๐‘ฅ [mm] turns 0.34 1 2 0.338 1 0.5 0.336 vertical tune ๐œˆ ๐‘ฆ ๐‘ [mrad] 0.334 0 ๐‘ NF,2 0 0.332 -1 -0.5 0.33 0.328 -2 -1 0.326 -40 -20 0 20 40 -1 -0.5 0 0.5 1 0.935 0.94 0.945 0.95 ๐‘ฆ [mm] ๐‘ž NF,2 radial tune ๐œˆ๐‘ฅ Figure 5.16: The vertical phase space behavior of this particle (๐›ฟ ๐‘ = 0.196%) has a slight triangular deformation. The overall normal form radius indicates a modulated amplitude and the spread out tune footprint starts right after the vertical 1/3-resonance line. Despite slight influence of the resonance, the rather elliptical phase space behavior makes this a trivial large amplitude loss. 120 1.1 6 40 1 normal form radius ๐‘Ÿ NF 4 20 0.9 2 ๐‘Ž [mrad] ๐‘ฆ [mm] 0.8 0 0 0.7 -2 -20 0.6 -4 0.5 -40 -6 0.4 -40 -20 0 20 40 -40 -20 0 20 40 1000 2000 3000 4000 ๐‘ฅ [mm] ๐‘ฅ [mm] turns 0.34 1 2 0.338 1 0.5 0.336 vertical tune ๐œˆ ๐‘ฆ ๐‘ [mrad] 0.334 0 ๐‘ NF,2 0 0.332 -1 -0.5 0.33 0.328 -2 -1 0.326 -40 -20 0 20 40 -1 -0.5 0 0.5 1 0.935 0.94 0.945 0.95 ๐‘ฆ [mm] ๐‘ž NF,2 radial tune ๐œˆ๐‘ฅ Figure 5.17: This particle (๐›ฟ ๐‘ = โˆ’0.088%) is caught around a period-3 fixed point structure in the vertical phase space, which is related to the vertical 1/3-resonance. We refer to these structures as islands and the loss mechanisms is called island related loss. 121 1.1 6 40 1 normal form radius ๐‘Ÿ NF 4 20 0.9 2 ๐‘Ž [mrad] ๐‘ฆ [mm] 0.8 0 0 0.7 -2 -20 0.6 -4 0.5 -40 -6 0.4 -40 -20 0 20 40 -40 -20 0 20 40 1000 2000 3000 4000 ๐‘ฅ [mm] ๐‘ฅ [mm] turns 0.34 1 2 0.338 1 0.5 0.336 vertical tune ๐œˆ ๐‘ฆ ๐‘ [mrad] 0.334 0 ๐‘ NF,2 0 0.332 -1 -0.5 0.33 0.328 -2 -1 0.326 -40 -20 0 20 40 -1 -0.5 0 0.5 1 0.935 0.94 0.945 0.95 ๐‘ฆ [mm] ๐‘ž NF,2 radial tune ๐œˆ๐‘ฅ Figure 5.18: This particle (๐›ฟ ๐‘ = โˆ’0.015%) forms large islands around a period-3 fixed point structure in the vertical phase space, which is associated with a major modulation of the oscillation amplitude. 122 1.1 6 40 1 normal form radius ๐‘Ÿ NF 4 20 0.9 2 ๐‘Ž [mrad] ๐‘ฆ [mm] 0.8 0 0 0.7 -2 -20 0.6 -4 0.5 -40 -6 0.4 -40 -20 0 20 40 -40 -20 0 20 40 1000 2000 3000 4000 ๐‘ฅ [mm] ๐‘ฅ [mm] turns 0.34 1 2 0.338 1 0.5 0.336 vertical tune ๐œˆ ๐‘ฆ ๐‘ [mrad] 0.334 0 ๐‘ NF,2 0 0.332 -1 -0.5 0.33 0.328 -2 -1 0.326 -40 -20 0 20 40 -1 -0.5 0 0.5 1 0.935 0.94 0.945 0.95 ๐‘ฆ [mm] ๐‘ž NF,2 radial tune ๐œˆ๐‘ฅ Figure 5.19: This particle (๐›ฟ ๐‘ = โˆ’0.127%) jumps between the islands. The large radial amplitude and/or the closeness to the (17/18, 1/3) resonance point might have triggered the jump. This is an example of moderate unstable behavior around a period-3 fixed point structure. 123 1.1 6 40 1 normal form radius ๐‘Ÿ NF 4 20 0.9 2 ๐‘Ž [mrad] ๐‘ฆ [mm] 0.8 0 0 0.7 -2 -20 0.6 -4 0.5 -40 -6 0.4 -40 -20 0 20 40 -40 -20 0 20 40 1000 2000 3000 4000 ๐‘ฅ [mm] ๐‘ฅ [mm] turns 0.34 1 2 0.338 1 0.5 0.336 vertical tune ๐œˆ ๐‘ฆ ๐‘ [mrad] 0.334 0 ๐‘ NF,2 0 0.332 -1 -0.5 0.33 0.328 -2 -1 0.326 -40 -20 0 20 40 -1 -0.5 0 0.5 1 0.935 0.94 0.945 0.95 ๐‘ฆ [mm] ๐‘ž NF,2 radial tune ๐œˆ๐‘ฅ Figure 5.20: This particle (๐›ฟ ๐‘ = 0.024%) shows a different kind of moderate unstable behavior around a period-3 fixed point structure, where the island size varies. The particle has both, a large radial amplitude and the closeness to the (17/18, 1/3) resonance point. 124 1.1 6 40 1 normal form radius ๐‘Ÿ NF 4 20 0.9 2 ๐‘Ž [mrad] ๐‘ฆ [mm] 0.8 0 0 0.7 -2 -20 0.6 -4 0.5 -40 -6 0.4 -40 -20 0 20 40 -40 -20 0 20 40 1000 2000 3000 4000 ๐‘ฅ [mm] ๐‘ฅ [mm] turns 0.34 1 2 0.338 1 0.5 0.336 vertical tune ๐œˆ ๐‘ฆ ๐‘ [mrad] 0.334 0 ๐‘ NF,2 0 0.332 -1 -0.5 0.33 0.328 -2 -1 0.326 -40 -20 0 20 40 -1 -0.5 0 0.5 1 0.935 0.94 0.945 0.95 ๐‘ฆ [mm] ๐‘ž NF,2 radial tune ๐œˆ๐‘ฅ Figure 5.21: This particle (๐›ฟ ๐‘ = 0.140%) forms a shuriken like shape in the vertical phase space. In this pattern there are two period-3 fixed point structures involved indicated by the double crossing of the vertical 1/3 resonance line. 125 1.1 6 40 1 normal form radius ๐‘Ÿ NF 4 20 0.9 2 ๐‘Ž [mrad] ๐‘ฆ [mm] 0.8 0 0 0.7 -2 -20 0.6 -4 0.5 -40 -6 0.4 -40 -20 0 20 40 -40 -20 0 20 40 1000 2000 3000 4000 ๐‘ฅ [mm] ๐‘ฅ [mm] turns 0.34 1 2 0.338 1 0.5 0.336 vertical tune ๐œˆ ๐‘ฆ ๐‘ [mrad] 0.334 0 ๐‘ NF,2 0 0.332 -1 -0.5 0.33 0.328 -2 -1 0.326 -40 -20 0 20 40 -1 -0.5 0 0.5 1 0.935 0.94 0.945 0.95 ๐‘ฆ [mm] ๐‘ž NF,2 radial tune ๐œˆ๐‘ฅ Figure 5.22: This particle (๐›ฟ ๐‘ = 0.196%) illustrates moderate unstable behavior in a shuriken pattern. The radial amplitude is not particularly large, but the resonance point (17/18, 1/3) is very close, which might be the trigger of the unsuitability. The unstable behavior is also visible in the continuously increasing normal form radius. 126 1.1 6 40 1 normal form radius ๐‘Ÿ NF 4 20 0.9 2 ๐‘Ž [mrad] ๐‘ฆ [mm] 0.8 0 0 0.7 -2 -20 0.6 -4 0.5 -40 -6 0.4 -40 -20 0 20 40 -40 -20 0 20 40 1000 2000 3000 4000 ๐‘ฅ [mm] ๐‘ฅ [mm] turns 0.34 1 2 0.338 1 0.5 0.336 vertical tune ๐œˆ ๐‘ฆ ๐‘ [mrad] 0.334 0 ๐‘ NF,2 0 0.332 -1 -0.5 0.33 0.328 -2 -1 0.326 -40 -20 0 20 40 -1 -0.5 0 0.5 1 0.935 0.94 0.945 0.95 ๐‘ฆ [mm] ๐‘ž NF,2 radial tune ๐œˆ๐‘ฅ Figure 5.23: This particle (๐›ฟ ๐‘ = 0.242%) illustrates a shuriken pattern, where the two period-3 fixed point structures are more obvious. The muon experiences a major modulation in the vertical oscillation amplitude and performs a double crossing of the vertical 1/3 resonance line. 127 1.1 6 40 1 normal form radius ๐‘Ÿ NF 4 20 0.9 2 ๐‘Ž [mrad] ๐‘ฆ [mm] 0.8 0 0 0.7 -2 -20 0.6 -4 0.5 -40 -6 0.4 -40 -20 0 20 40 -40 -20 0 20 40 1000 2000 3000 4000 ๐‘ฅ [mm] ๐‘ฅ [mm] turns 0.34 1 2 0.338 1 0.5 0.336 vertical tune ๐œˆ ๐‘ฆ ๐‘ [mrad] 0.334 0 ๐‘ NF,2 0 0.332 -1 -0.5 0.33 0.328 -2 -1 0.326 -40 -20 0 20 40 -1 -0.5 0 0.5 1 0.935 0.94 0.945 0.95 ๐‘ฆ [mm] ๐‘ž NF,2 radial tune ๐œˆ๐‘ฅ Figure 5.24: This particle (๐›ฟ ๐‘ = โˆ’0.096%) shows a shuriken pattern with unstable tendencies. The large radial amplitude and/or the closeness to the radial 17/18 resonance line might be the trigger for the instability. 128 1.1 6 40 1 normal form radius ๐‘Ÿ NF 4 20 0.9 2 ๐‘Ž [mrad] ๐‘ฆ [mm] 0.8 0 0 0.7 -2 -20 0.6 -4 0.5 -40 -6 0.4 -40 -20 0 20 40 -40 -20 0 20 40 1000 2000 3000 4000 ๐‘ฅ [mm] ๐‘ฅ [mm] turns 0.34 1 2 0.338 1 0.5 0.336 vertical tune ๐œˆ ๐‘ฆ ๐‘ [mrad] 0.334 0 ๐‘ NF,2 0 0.332 -1 -0.5 0.33 0.328 -2 -1 0.326 -40 -20 0 20 40 -1 -0.5 0 0.5 1 0.935 0.94 0.945 0.95 ๐‘ฆ [mm] ๐‘ž NF,2 radial tune ๐œˆ๐‘ฅ Figure 5.25: This particle (๐›ฟ ๐‘ = โˆ’0.159%) shows a shuriken pattern with a moderate instability. The two period-3 fixed point structures are so close together that the particle gets temporarily caught around the inner one of them in an island pattern. 129 1.1 6 40 1 normal form radius ๐‘Ÿ NF 4 20 0.9 2 ๐‘Ž [mrad] ๐‘ฆ [mm] 0.8 0 0 0.7 -2 -20 0.6 -4 0.5 -40 -6 0.4 -40 -20 0 20 40 -40 -20 0 20 40 1000 2000 3000 4000 ๐‘ฅ [mm] ๐‘ฅ [mm] turns 0.34 1 2 0.338 1 0.5 0.336 vertical tune ๐œˆ ๐‘ฆ ๐‘ [mrad] 0.334 0 ๐‘ NF,2 0 0.332 -1 -0.5 0.33 0.328 -2 -1 0.326 -40 -20 0 20 40 -1 -0.5 0 0.5 1 0.935 0.94 0.945 0.95 ๐‘ฆ [mm] ๐‘ž NF,2 radial tune ๐œˆ๐‘ฅ Figure 5.26: This particle (๐›ฟ ๐‘ = 0.181%) shows the pattern of a very blunt shuriken. The vertical amplitude oscillation is only moderate and illustrates there can be almost regular behavior between two period-3 fixed point structures. 130 1.1 6 40 1 normal form radius ๐‘Ÿ NF 4 20 0.9 2 ๐‘Ž [mrad] ๐‘ฆ [mm] 0.8 0 0 0.7 -2 -20 0.6 -4 0.5 -40 -6 0.4 -40 -20 0 20 40 -40 -20 0 20 40 1000 2000 3000 4000 ๐‘ฅ [mm] ๐‘ฅ [mm] turns 0.34 1 2 0.338 1 0.5 0.336 vertical tune ๐œˆ ๐‘ฆ ๐‘ [mrad] 0.334 0 ๐‘ NF,2 0 0.332 -1 -0.5 0.33 0.328 -2 -1 0.326 -40 -20 0 20 40 -1 -0.5 0 0.5 1 0.935 0.94 0.945 0.95 ๐‘ฆ [mm] ๐‘ž NF,2 radial tune ๐œˆ๐‘ฅ Figure 5.27: This particle (๐›ฟ ๐‘ = 0.106%) is characterized by a very large vertical amplitude, which is additionally modulated by the shuriken pattern. Its one of the very few particles for which the orbit considerably overlaps with the collimator boundary. 131 1.1 6 40 1 normal form radius ๐‘Ÿ NF 4 20 0.9 2 ๐‘Ž [mrad] ๐‘ฆ [mm] 0.8 0 0 0.7 -2 -20 0.6 -4 0.5 -40 -6 0.4 -40 -20 0 20 40 -40 -20 0 20 40 1000 2000 3000 4000 ๐‘ฅ [mm] ๐‘ฅ [mm] turns 0.34 1 2 0.338 1 0.5 0.336 vertical tune ๐œˆ ๐‘ฆ ๐‘ [mrad] 0.334 0 ๐‘ NF,2 0 0.332 -1 -0.5 0.33 0.328 -2 -1 0.326 -40 -20 0 20 40 -1 -0.5 0 0.5 1 0.935 0.94 0.945 0.95 ๐‘ฆ [mm] ๐‘ž NF,2 radial tune ๐œˆ๐‘ฅ Figure 5.28: This particle (๐›ฟ ๐‘ = 0.118%) shows strong instabilities caused by a combination of a very large vertical amplitude in combination with a period-3 fixed point structure, which occasionally captures the orbit in an island pattern. 132 1.1 6 40 1 normal form radius ๐‘Ÿ NF 4 20 0.9 2 ๐‘Ž [mrad] ๐‘ฆ [mm] 0.8 0 0 0.7 -2 -20 0.6 -4 0.5 -40 -6 0.4 -40 -20 0 20 40 -40 -20 0 20 40 1000 2000 3000 4000 ๐‘ฅ [mm] ๐‘ฅ [mm] turns 0.34 1 2 0.338 1 0.5 0.336 vertical tune ๐œˆ ๐‘ฆ ๐‘ [mrad] 0.334 0 ๐‘ NF,2 0 0.332 -1 -0.5 0.33 0.328 -2 -1 0.326 -40 -20 0 20 40 -1 -0.5 0 0.5 1 0.935 0.94 0.945 0.95 ๐‘ฆ [mm] ๐‘ž NF,2 radial tune ๐œˆ๐‘ฅ Figure 5.29: This particle (๐›ฟ ๐‘ = 0.010%) diverges due to its unstable orbit. The approach of the unstable fixed point with such a large vertical amplitude are likely the trigger of the divergence. 133 Another property that many lost particles share is a significant momentum offset, which radially shifts their respective reference orbit closer to the boundaries of the collimator. The dependence of the radial position of the reference orbit on the momentum offset decreases the maximum survivable size of those rectangular shapes in ๐‘ฅ๐‘ฆ space significantly as previously discussed in Sec. 5.3.3 and illustrated in Fig. 5.4. Last but not least, there are also particles like the one shown in Fig. 5.15, which get lost simply because of their constant but large oscillation amplitudes in the radial and/or vertical direction. However, it is not always obvious to distinguish them from particles that are under the influence of a period-3 fixed point structure like the particle in Fig. 5.16. Fig. 5.15 to Fig. 5.29 also indicate that the ๐‘ฅ๐‘ฆ pattern of lost particles often only barely touches the collimator boundary. For these cases, it may take many revolutions for both oscillations, in the radial and vertical direction, to reach their maximum simultaneously [84]. 5.5.3 Period-3 Fixed Point Structures There are period-3 fixed point structures in the vertical phase space as seen, for example, in Fig. 5.17. The period-3 fixed points are a property of the vertical projection of the stroboscopic muon tracking. They are associated with the vertical 1/3-resonance, which is particularly relevant due to the strong eight order nonlinear tune shifts from the strong ninth order nonlinear field contributions of the 20th order multipole of the potential from the ESQ [82]. The period-3 fixed point structure corresponds to an orbit, which vertically oscillates around its momentum dependent reference orbit with a period of exactly three turns, i.e. a vertical betatron tune of 1/3. However, such an orbit is not necessarily a closed orbit, which closes after three turns, because while the vertical behavior might be exactly resonant after three turns, the radial behavior is not. There are stable fixed points and unstable fixed points within the period-3 fixed point structures. Accordingly, the term โ€˜period-3 fixed pointsโ€™ describes a set of 6 fixed points at the same amplitude in ๐‘ฆ๐‘, where every other fixed point is stable. The positions of the period-3 fixed points in the 134 vertical phase space depend on the momentum offset ๐›ฟ ๐‘ and the radial phase space (due to coupling). The combination of stable and unstable fixed points creates island patterns around the stable fixed points as shown in Fig. 5.30. 2 1 b [mrad] 0 โˆ’1 โˆ’2 โˆ’40 โˆ’20 0 20 40 y [mm] Figure 5.30: The left plot shows stroboscopic tracking in the vertical phase space illustrating orbit behavior with a single period-3 fixed point structure present. The orbits only differ in their vertical phase space behavior โ€“ they all have the same momentum offset of ๐›ฟ ๐‘ = 0.126 % and are at the momentum dependent equilibrium point in radial phase space (๐‘ฅ = 10.64 mm, ๐‘Ž = 0.045 mrad) having no radial oscillation amplitude. The blue orbits indicate the island patterns around the stable fixed points in the middle of the islands. The red orbits are right at the edge before being caught around the fixed points. The three unstable fixed points are in the space between the two red orbits, where the islands almost touch. In the right plot, the attractive (green) and repulsive (violet) eigenvectors of the linear dynamics around the unstable fixed points are schematically shown. The unstable fixed points are located in the blank space between the islands. The linear dynamics around them are characterized by an attractive eigenvector with a corresponding eigenvalue smaller than one and a repelling eigenvector with an eigenvalue larger than one. Those unstable fixed points give rise to chaotic behavior because two phase space orbits that are initially near yield widely diverging dynamical behavior from each other once they come close to the unstable fixed point. The 135 inner red orbit and the adjacent blue island orbit in Fig. 5.30 illustrate this chaotic behavior. They both approach the unstable fixed points along the attractive eigenvector (green), but the unstable eigenvector (violet) ejects them in opposite directions. While the inner red orbit appears almost regular like the black orbits of lower amplitudes, the muon on the blue orbit with a slightly larger amplitude gets ejected outwards by the unstable fixed point, which drastically increases its vertical amplitude. In the case shown in Fig. 5.30, the stable fixed point is able to keep the particle in an island orbit. In Fig. 5.29, on the other hand, the particle cannot remain on the island orbit and diverges. In [98, 99], a similar analysis of the accelerator transfer map representing the Tevatron is performed and rigorous methods to determine the position of those fixed point structures are presented. It is also not uncommon for two period-3 fixed point structures to be present simultaneously in the vertical phase space. Often the structures are oriented such that a stable fixed point of structure with the larger amplitude is โ€˜aboveโ€™ an unstable fixed point of the structure with the lower amplitude. In Fig. 5.31 a phase space region with two period-3 fixed point structures for orbits with ๐›ฟ ๐‘ = 0.339 % is shown. The different plots illustrate how the relative position and interaction of the two period-3 fixed point structures drastically changes the dynamics for particles that are initiated on initially very near orbits. This further emphasizes the potential to chaotic behavior caused by the unstable fixed points within those period-3 fixed point structures. The two period-3 fixed point structures can be well separated, as shown in a), yielding the known island patterns with โ€˜regularโ€™ orbits in between. However, the structures can also move into each other such that some orbits are caught between the two period-3 fixed point structures and follow the shape of a threefold shuriken around the two island patterns as shown in green in b) and c). When the two period-3 fixed point structures come even closer, the opposite fixed points of the two period-3 fixed point structures can annihilate each other, resulting in triangular patterns with rounded corners (see gray patterns in d)). 136 1.5 a) ๐‘ฅ 1.5 b) ๐‘ฅ amp = 6 mm amp = 4.8 mm 1 1 0.5 0.5 b [mrad] b [mrad] 0 0 -0.5 -0.5 -1 -1 -1.5 -1.5 -30 -20 -10 0 10 20 30 -30 -20 -10 0 10 20 30 y [mm] y [mm] 1.5 c) ๐‘ฅ 1.5 d) ๐‘ฅ amp = 4 mm amp = 1 mm 1 1 0.5 0.5 b [mrad] b [mrad] 0 0 -0.5 -0.5 -1 -1 -1.5 -1.5 -30 -20 -10 0 10 20 30 -30 -20 -10 0 10 20 30 y [mm] y [mm] Figure 5.31: Stroboscopic tracking in the vertical phase space illustrating orbit behavior with two period-3 fixed point structures present. The orbits in each plot only differ in their vertical phase space behavior. All orbits have the same momentum offset of ๐›ฟ ๐‘ = 0.339 %. The four plots differ by their radial amplitude around the momentum dependent equilibrium point in radial phase space at (๐‘ฅ = 27.7 mm, ๐‘Ž = 0.144 mrad). The radial amplitudes are: a) ๐‘ฅ amp = 6 mm, b) ๐‘ฅamp = 4.8 mm, c) ๐‘ฅ amp = 4 mm, d) ๐‘ฅ amp = 1 mm. The blue orbits indicate the island patterns around the stable fixed points. The red orbits are right at the edge before being caught around the period-3 fixed points. The green orbits are caught around both period-3 fixed point structures. The gray orbits in d) emphasize that half of the fixed points from c) have indeed been annihilated. 137 While the period-3 fixed point structures often lead to a significant vertical amplitude modulation, many of them are well within the boundary of the collimators like the examples shown in Fig. 5.30 and Fig. 5.31. So, the involvement of a particle in a period-3 fixed point structure or two does not necessarily mean that it is lost, but the additional modulation of the vertical amplitude drastically increases the risk of getting lost for those particles. All orbit patterns shown in Fig. 5.15 to Fig. 5.29 can be found in a similar form either in Fig. 5.30 or Fig. 5.31. In other words, we fully understand what is causing the different types of patterns. The major difference for some particles is the stability of their pattern. The phase space regions chosen in Fig. 5.30 and Fig. 5.31 are stable and do not share the characteristics of unstable orbits which are large radial amplitudes and/or closeness to the 17/18 resonance point. 5.5.4 Muon Loss Rates from Simulation We have seen what different phase space tracking patterns can arise due to period-3 fixed point structures. We also saw that these structures can be responsible for losses due to the modulation of the oscillation amplitude in the vertical phase space. To get a more general understanding of how prominent these patterns are among the entire distribution and how common they are among lost particles, we need a mechanism to characterize these patterns in a way that can be automatically detected. The various degrees of instabilities, especially among particles involved with period-3 fixed point structures make a generalized categorization difficult. There is no obvious distinction between certain unstable islands and certain shuriken patterns, and also no clear distinction between very blunt shuriken patterns and very triangularly deformed regular elliptical patterns. Accordingly, we only make two distinctions. First, we try to distinguish between particles involved with the vertical 1/3-resonance and particles that are not. Among the particles that are involved with the vertical 1/3-resonance, we make a further distinction between pure island patterns and everything else. A pure island pattern is a (non-across-jumping) island pattern. Fig. 5.19 shows an across-jumping island structure, where the orbit jumps from one fixed point island to another. In comparison, 138 Fig. 5.20 also shows an unstable island pattern, but one that remains on the island around the fixed points. For reference, we will call particles that we detect as being involved with the vertical 1/3- resonance โ€˜period-3 particlesโ€™ and all the others โ€˜regular particlesโ€™. Of the period-3 particles, we will only give a special name to the โ€˜island particlesโ€™, because the period-3 non-island particles are a very diverse group, which is not easily described by a single word without mischaracterizing at least some of its elements. Since the transition between patterns is continuous as the gray orbits in Fig. 5.31d illustrates, the category of period-3 particles and the category of regular particles might have elements that are almost identical. The following paragraph clarifies how we define the different categories with our detection mechanisms. We start by explaining how we identify period-3 particles. We consider the vertical phase space in polar coordinates and look at the phase space behavior in steps of three. The first three vertical phase space angles during tracking are denoted by ๐œ™1,1 , ๐œ™2,1 and ๐œ™3,1 , and the next three angles are denoted by ๐œ™1,2 , ๐œ™2,2 and ๐œ™3, . Starting from the initial angle ๐œ™1,1 , we have angles after each turn as ๐œ™2,1 , ๐œ™3,1 , ๐œ™1,2 , ๐œ™2,2 , ๐œ™3,2 , ..., ๐œ™1,๐‘› , ๐œ™2,๐‘› , ๐œ™3,๐‘› until the total turn number is reached. For our tracking, its 4500 turns in total, which corresponds to ๐‘› = 1500. Additionally, we define the angle advances ฮ”๐œ™๐‘–,๐‘› = ๐œ™๐‘–,๐‘› โˆ’ ๐œ™๐‘–,๐‘›โˆ’1 . To avoid ambiguity in the value for the angles,   we require that value for ๐œ™๐‘–,๐‘› is chosen such that ๐œ™๐‘–,๐‘› โˆˆ ๐œ™๐‘–,๐‘›โˆ’1 โˆ’ ๐œ‹, ๐œ™๐‘–,๐‘›โˆ’1 + ๐œ‹ . If there is a sign change from ฮ”๐œ™๐‘–,๐‘›โˆ’1 to ฮ”๐œ™๐‘–,๐‘› the sign-change-count ๐œ…๐‘– is increased by one. If all ๐œ…๐‘– are nonzero after the 4500 turns, then we categorize the particle as period-3 particle. To identify island particles, we use the definitions from above and additionally introduce the   range D๐‘–,1 = ๐œ™๐‘–,1 , ๐œ™๐‘–,1 of the angles for each of the three potential island locations. With every iteration step the ranges of the angles are updated to      D๐‘–,๐‘› = D๐‘–,๐‘›,LB , D๐‘–,๐‘›,UB = min ๐œ™๐‘–,๐‘› , D๐‘–,๐‘›โˆ’1,LB , max ๐œ™๐‘–,๐‘› , D๐‘–,๐‘›โˆ’1,UB ) . (5.6) The abbreviations โ€˜LBโ€™ and โ€˜UBโ€™ denote the lower and upper bound of the range respectively. Note that the rule to avoid ambiguity in the value for the angles from above also applies here. All 139 particles for which the sum of the three ranges is less than a full revolution (2๐œ‹) after the 4500 tracking turns are considered island particles. In other words, island particles satisfy 3 โˆ‘๏ธ |D๐‘–,1500 | < 2๐œ‹. (5.7) ๐‘–=1 With these recognition mechanisms implemented, we were able to characterize all particles and determine their proportion as presented in Tab. 5.1. Period-3 particles are dominating among lost particles. Accordingly, period-3 particles and island particles, in particular, are more prone to be lost. But by far not every period-3 particle or island particle is lost. More than 77% of island particles and more than 92% of period-3 particles survive the 4500 turns. As Fig. 5.30 illustrates, sometimes the amplitude of these period-3 structures is so low that the additional modulation of the amplitude is not enough to be critical. Table 5.1: Percentages of different characterization groups. Read as follows: ๐‘ฅ % of Base particles have the property Property. All particles that hit a collimator during the 4500 turns of tracking are considered lost. XXX XXX Base XXX All Lost Period-3 Island Property XXX Lost 0.686% 100% 7.44% 22.2% Period-3 7.06% 76.4% 100% 100% Island 1.00% 32.4% 14.2% 100% While island particles make up only 1/7 of period-3 particles, they are responsible for almost half the losses associated with period-3 particles. This is particularly surprising because the island particle category excludes most unstable patterns by definition (exceptions are moderate instabilities that do not contravene the recognition criteria like the particle shown in Fig. 5.20). On the other hand, period-3 particles cover a wide range of patterns some of which barely show a modulation of the vertical amplitude as the example of the gray orbits in Fig. 5.31d shows. To understand how the losses occur over time, we plot the accumulative loss ratio over the 4500 turns in Fig. 5.32. Island loss is the fastest growing loss over the first 1000 turns before settling almost asymptotically. This is explained by different modulation frequencies around the period-3 fixed point structures. The closer to the unstable fixed point, the larger the modulation and the 140 slower the modulation frequency. Accordingly, the island modulation is on average faster than the shuriken modulation. turn turn 0 1000 2000 3000 4000 0 1000 2000 3000 4000 0.7 0.7 a) b) regular period-3 0.6 0.6 islands period-3 without islands cummulative losses [%] cummulative losses [%] 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 regular 0.1 period-3 islands 0 0 250 30 100 400 550 700 30 100 250 400 550 700 time since injection [๐œ‡s] time since injection [๐œ‡s] Figure 5.32: a) Shows how the muon loss ratio is composed of regular particles (purple) and particles involved with period-3 fixed point structures (green). Of the period-3 particles (green), the fraction caught in islands structures is indicated by the blue stripe pattern. In b) the loss ratio over time is shown for each subgroup of lost particles to better understand which losses drive to overall loss from plot a). The tracking starts after the initial 30 ๐œ‡s of final beam preparation when data taking is initiated. 5.6 Conclusion The Poincarรฉ return map description of the storage ring model of the Muon ๐‘”-2 Experiment [85] and its analysis with DA normal form methods yielded many insightful characteristics of the system. We gained an understanding of the form of the closed orbit within the storage ring as well as details on how it changes with an offset in the momentum ๐›ฟ ๐‘. Considering that particles oscillate around their corresponding reference orbit, which is the closed orbit of their momentum offset, the radial shift of the closed orbit with momentum offset is particularly critical. This shift brings the equilibrium state of the radial oscillation closer to the collimator boundary, which increases the risk 141 of particles getting lost. The tune analysis provided a detailed understanding of how the oscillation frequencies of particles depend on their momentum offset and their amplitudes relative to their respective reference orbit. This analysis showed that particles over the entire momentum offset range could cross the vertical 1/3-resonance frequency for certain vertical and radial amplitude combinations. The strong ninth order nonlinearities of the map caused by the 20th-pole of the ESQ potential have a significant effect on amplitude and parameter dependent tune shifts. This property manifests itself in the dominating eighth order dependencies in the amplitude and momentum dependent tune shifts and the drastic change in the tune footprint for calculations of order ๐‘š > 8, which include the ninth order terms of the original map. Further tracking analysis revealed period-3 fixed point structures in the vertical phase space. They are associated with the vertical 1/3-resonance tune and cause significant vertical amplitude modulations to the particles that are caught around them. We were able to connect all vertical phase space patterns of lost particles either with regular distorted elliptical patterns or with patterns that arise around one or two of these period-3 fixed point structures. Additionally, instabilities caused by large radial amplitudes and/or closeness to the (17/18) resonance point significantly mixed multiple of the known orbit patterns. This only allowed for a limited automatic recognition of patterns, which in turn revealed valuable insights about the effect of these period-3 fixed point structures on the loss rates of muons in the storage ring. Particles associated with period-3 fixed point structures are at a higher risk of getting lost. 142 CHAPTER 6 VERIFIED CALCULATIONS USING TAYLOR MODELS In this chapter, we take steps towards making the methods presented above self-verified. Since many aspects that have to be carefully considered for a rigorous transfer to the verified world lay beyond the scope of this thesis, this chapter will only yield a discussion of the basic principles behind some of them. However, the aspect of verified global optimization and its application to the normal form defect for verified stability estimates will be analyzed in greater detail. To introduce the concept of verified global optimization using Taylor Models (TM), we first apply it to two well-known example optimization problems. First, in Sec. 6.1, we run a Taylor Model based verified global optimization in different operating modes on the 2D and the generalized Rosenbrock function, as it is one of the most commonly used examples to test global optimization algorithms. In Sec. 6.2, we discuss the optimization problem of finding minimum energy configurations of particles that have their pairwise interaction energy modeled by the Lennard-Jones potential. It is one of the simplest examples to explain, yet arbitrarily complex to solve depending on the number of particles in the configuration and the dimensionality of the configuration. In comparison to the Rosenbrook example, the setup of the Lennard-Jones optimization problem is more complex especially for configuration in 2D and 3D. In Sec. 6.3, we discuss the intricacies of using the methods from Chapter 4 and Chapter 5 for a verified stability analysis of those dynamical systems. In particular, we take a detailed look at options of utilizing the normal form defect from Sec. 2.4 as a measure of stability. With the gained understanding of verified global optimization from Sec. 6.1 and Sec. 6.2, we analyze its application to the normal form defect to calculate verified stability estimates for the simulated phase space behavior in the Muon ๐‘”-2 Storage Ring. 143 6.1 The Rosenbrock Optimization Problem 6.1.1 The Rosenbrock Function The Rosenbrock function  2 ๐‘“ (๐‘ฅ, ๐‘ฆ) = (๐‘Ž โˆ’ ๐‘ฅ) 2 +๐‘ ๐‘ฆ โˆ’ ๐‘ฅ2 (6.1) was introduced by Howard H. Rosenbrock [78]. It is a non-convex function that is commonly used as a test problem for optimization algorithms. The parameters are usually set to (๐‘Ž, ๐‘) = (1, 100), and so we will use those parameters here as well. Fig. 6.1 illustrates the Rosenbrock function for those parameters. 1.5 1000 1 100 0.5 10 ๐‘ฆ 0 1 โˆ’0.5 0.1 โˆ’1 โˆ’1.5 0.01 โˆ’1.5 โˆ’1 โˆ’0.5 0 0.5 1 1.5 ๐‘ฅ Figure 6.1: A contour plot of the Rosenbrock function with (๐‘Ž, ๐‘) = (1, 100). The Rosenbrock function is also referred to as Rosenbrockโ€™s valley function or Rosenbrockโ€™s banana function for obvious reasons. It is characterized by a long and deep valley, the floor of which constitutes a shallow valley. This shallowness is one of the aspects that challenges optimizers. 144 There are various multidimensional generalizations of the Rosenbrock function to compare more advanced optimization algorithms. In this work, we will use the following generalized form ๐‘›โˆ’1 โˆ‘๏ธ   2  ๐‘“๐‘›D ( ๐‘ฅยฎ) = 100 ๐‘ฅ๐‘–+1 โˆ’ ๐‘ฅ๐‘–2 + (1 โˆ’ ๐‘ฅ๐‘– )2 , (6.2) ๐‘–=1 where ๐‘› โ‰ฅ 2 is the dimension and ๐‘ฅ๐‘– are the optimization variables. Note that this generalized definition is consistent with the definition of the 2D Rosenbrock function from above and also retains the difficulties of the original problem of a deep valley with a long and shallow valley floor, but with a complexity that increases with ๐‘›. Unless specified otherwise, we will refer to the generalized Rosenbrock function as the Rosenbrock function or the objective function of the optimization. The Rosenbrock function is a composition of quadratic expressions. Because of the double squares, the Rosenbrock function is always a fourth order polynomial. Additionally, none of the individual terms in the sum can be negative. Hence, a global minimum would be reached if all individual terms of the sum are zero. The (1 โˆ’ ๐‘ฅ๐‘– ) 2 terms are only zero for ๐‘ฅ๐‘– = 1, which also yields zero for the remaining terms. Accordingly, ๐‘ฅยฎโ˜… = (1, 1, ..., 1) is the single global minimum of the Rosenbrock function for which every term is zero and therefore the overall objective function is zero. In Fig. 6.2, the Rosenbrock function is illustrated in multiple 2D projections around its minimum at ๐‘ฅยฎโ˜…. In other words, all ๐‘ฅ๐‘– are set to one except for the variables shown in the projection. The Rosenbrock function also has a dependency problem. For the first variable ๐‘ฅ 1 , the following dependent terms appear  2 100 ๐‘ฅ 2 โˆ’ ๐‘ฅ 12 + (1 โˆ’ ๐‘ฅ1 ) 2 . (6.3) For any of the variables ๐‘ฅ๐‘– with 1 < ๐‘– < ๐‘›, there is one additional dependent term with  2  2 100 ๐‘ฅ๐‘–+1 โˆ’ ๐‘ฅ๐‘–2 + (1 โˆ’ ๐‘ฅ๐‘– ) 2 + 100 ๐‘ฅ๐‘– โˆ’ ๐‘ฅ๐‘–โˆ’1 2 . (6.4) The last variable ๐‘ฅ ๐‘› , only appears in one term, namely  2 2 100 ๐‘ฅ ๐‘› โˆ’ ๐‘ฅ ๐‘›โˆ’1 . (6.5) 145 1.5 1.5 1 1 ๐‘ฅ๐‘–+1 for 1 < ๐‘– < ๐‘› โˆ’ 1 0.5 0.5 ๐‘ฅ2 0 0 โˆ’0.5 โˆ’0.5 โˆ’1 โˆ’1 โˆ’1.5 โˆ’1.5 โˆ’1.5 โˆ’1 โˆ’0.5 0 0.5 1 1.5 โˆ’1.5 โˆ’1 โˆ’0.5 0 0.5 1 1.5 ๐‘ฅ1 ๐‘ฅ๐‘– for 1 < ๐‘– < ๐‘› โˆ’ 1 1.5 1.5 1 1 ๐‘ฅ ๐‘— for ๐‘– + 1 < ๐‘— < ๐‘› 0.5 0.5 0 ๐‘ฅ๐‘› 0 โˆ’0.5 โˆ’0.5 โˆ’1 โˆ’1 โˆ’1.5 โˆ’1.5 โˆ’1.5 โˆ’1 โˆ’0.5 0 0.5 1 1.5 โˆ’1.5 โˆ’1 โˆ’0.5 0 0.5 1 1.5 ๐‘ฅ๐‘– for 1 < ๐‘– < ๐‘› โˆ’ 2 ๐‘ฅ ๐‘›โˆ’1 Figure 6.2: Projections of the multidimensional generalizations of the Rosenbrock function (Eq. (6.2)) into 2D-subspaces around minimum at ๐‘ฅยฎ = (1, 1, ..., 1), i.e., all variables are equal one except for the ones shown in the respective plot. 6.1.2 Global Optimization Using COSY-GO The global optimization is performed using COSY-GO [63, 64, 59]. In the most advanced setting (QFB/LDB), the algorithm uses both of the advanced Taylor Model based bounding methods, namely, the quadratic fast bounder (QFB) and the linear dominated bounder (LDB), which were mentioned in Sec. 2.6 and were introduced in [64]. Additionally, COSY-GO also uses naive Taylor Model bounding and interval evaluations (IN). For comparisons, COSY-GO offers to run an optimization with some of the advanced methods disabled. By ranking the bounding methods in the order: QFB, 146 LDB, naive TM, and IN, the operating mode is denoted by its highest ranking bounding method, e.g., the running mode LDB indicates that LDB, naive TM, and IN are used but not QFB. Because the global minimum of the Rosenbrock function is already known, we are just interested in the algorithmโ€™s performance to narrow down the domain of the minimum and its value. Accordingly, we can choose an arbitrary search domain for the optimization that includes ๐‘ฅยฎโ˜…. We will investigate the Rosenbrock function over the domain [โˆ’1.5, 1.5] ๐‘› . For the optimization, we evaluate the objective function the way it is written in Eq. (6.2) and not expanded out in a single second, third, and fourth order polynomial terms. We also clarify that the optimization is performed with no additional knowledge about the derivatives of the objective function. 6.1.2.1 Illustration of the Cluster Effect and Dependency Problem using the 2D Rosenbrock Function In Fig. 6.3, the performance of COSY-GO on the 2D Rosenbrock function is visualized in the form of its splitting pattern. It shows the individual boxes analyzed in the various operation modes. All calculations are performed with fourth order Taylor Models (TM) except for the interval evaluation, which does not use TM. The significant differences in the splitting patterns are the number of splits, and the way boxes are split. For the operating mode in naive TM and IN, boxes are always split in half, where each of the two modes has its own methods of deciding in which variable domain the box is split, i.e., for 2D either splitting in ๐‘ฅ or in ๐‘ฆ. With LDB and QFB, the boxes are decreased in size as the respective method sees fit. Especially close to the minimum this avoids the cluster effect [44, 37]. In Fig. 6.4, the boxing close to the minimum is illustrated, which clearly shows the cluster effect and its avoidance using QFB/LDB. Another advantage of the Taylor Model based approach is the avoidance of the dependency problem [55]. However, due to the simplicity of the 2D Rosenbrock function and its weak dependency problem in the form from Eq. (6.2), the advantages of the Taylor Model based methods are not so 147 1.5 1.5 1 1 0.5 0.5 ๐‘ฆ 0 ๐‘ฆ 0 โˆ’0.5 โˆ’0.5 โˆ’1 โˆ’1 QFB/LDB LDB โˆ’1.5 โˆ’1.5 โˆ’1.5 โˆ’1 โˆ’0.5 0 0.5 1 1.5 โˆ’1.5 โˆ’1 โˆ’0.5 0 0.5 1 1.5 ๐‘ฅ ๐‘ฅ 1.5 1.5 1 1 0.5 0.5 ๐‘ฆ 0 ๐‘ฆ 0 โˆ’0.5 โˆ’0.5 โˆ’1 โˆ’1 naive TM IN โˆ’1.5 โˆ’1.5 โˆ’1.5 โˆ’1 โˆ’0.5 0 0.5 1 1.5 โˆ’1.5 โˆ’1 โˆ’0.5 0 0.5 1 1.5 ๐‘ฅ ๐‘ฅ Figure 6.3: Verified global optimization of the 2D Rosenbrock function using COSY-GO in different operation modes with fourth order Taylor Models for all modes except interval evaluations (IN). prominent relative to the IN evaluations. To still visually emphasize the advantages of the TM operations over intervals, we artificially increase the dependency problem in the objective function by modifying it to ๐‘“ = ๐‘“2D โˆ’ ๐‘“2D + ๐‘“2D . In Fig. 6.5, the QFB/LDB methods using fourth order Taylor Models are compared to the interval method for the modified objective function. Even though the fourth order TM representation of the modified objective function only differs from the TM representation of the non-modified objective function by a slightly different remainder bound, the behavior and efficiency of the algorithm with QFB/LDB change more for the modified 148 1.003 1.003 ๐‘ฆ 1 ๐‘ฆ 1 QFB/LDB LDB 0.997 0.997 0.997 1 1.003 0.997 1 1.003 ๐‘ฅ ๐‘ฅ 1.003 1.003 ๐‘ฆ 1 ๐‘ฆ 1 naive TM IN 0.997 0.997 0.997 1 1.003 0.997 1 1.003 ๐‘ฅ ๐‘ฅ Figure 6.4: No cluster effect for the COSY-GO operating mode QFB/LDB, but a significant cluster effect for the IN evaluation. objective function than one would initially expect. This is because the algorithm in the QFB/LDB mode also performs intermediate steps with lower order Taylor Models, which are quicker to evaluate but less accurate. Those lower order evaluations are more sensitive to the dependency problem, which explains the effect of those intermediate steps on the splitting decisions. 149 1.5 1.5 1 1 0.5 0.5 ๐‘ฆ 0 ๐‘ฆ 0 โˆ’0.5 โˆ’0.5 โˆ’1 โˆ’1 QFB/LDB ๐‘“ = ๐‘“2D โˆ’ ๐‘“2D + ๐‘“2D IN ๐‘“ = ๐‘“2D โˆ’ ๐‘“2D + ๐‘“2D โˆ’1.5 โˆ’1.5 โˆ’1.5 โˆ’1 โˆ’0.5 0 0.5 1 1.5 โˆ’1.5 โˆ’1 โˆ’0.5 0 0.5 1 1.5 ๐‘ฅ ๐‘ฅ Figure 6.5: Splitting comparison between fourth order Taylor Model approach with QFB/LDB enabled and interval evaluation using the example of the modified 2D Rosenbrock function. 6.1.2.2 Performance of COSY-GO for High Dimensional Rosenbrock Function Next, we analyze the performance of COSY-GO for the optimization of the higher dimensional Rosenbrock function in the form from Eq. (6.2). The search domain of the optimization is always set to [โˆ’1.5, 1.5] ๐‘› . Accordingly, the search volume increases exponentially with the dimension of the objective function. As a stopping condition of the algorithm, we require that boxes with a side length ๐‘  < 1E-6 are not split. Ideally, the optimizer reduces the search volume of 3๐‘› by at least a factor of 3,000,000๐‘› to a single box with a volume smaller than (1E-6) ๐‘› . In the most advanced setting (QFB/LDB), which requires a minimum Taylor Model order of two, COSY-GO manages to reduce the search domain to a single box with a side length ๐‘  < 1E-6 for every dimension ๐‘› that we tested. Fig. 6.6 illustrates how the performances of COSY-GO in the evaluation of the generalized Rosenbrock function from Eq. (6.2) varies for different Taylor Model orders in the QFB/LDB mode. For comparison, the performance of using interval evaluations is also shown. The second order calculation outperforms the higher order calculations in both aspects, regarding the speed and the number of required steps. On the one hand, the evaluation of higher order Taylor 150 100 105 10 time [s] 104 1 Steps QFB/LDB O2 103 QFB/LDB O3 0.1 QFB/LDB O4 QFB/LDB O5 IN 0.01 102 5 10 15 5 10 15 dimension ๐‘› dimension ๐‘› Figure 6.6: Time consumption and number of steps in the optimization of the regular ๐‘› dimensional Rosenbrock function from Eq. (6.2) at various orders with COSY-GO and QFB/LDB enabled. Additionally, the interval evaluation performance is also shown for comparison. Models takes longer than the evaluation of lower order Taylor Models. On the other hand, higher order Taylor Models usually provide tighter bounds, which decreases the number of required steps. Regarding the second aspect, the result from Fig. 6.6 on the required steps is rather unusual, because the low order Taylor Model of second order requires fewest steps than the higher order Taylor Models. This phenomenon is specific to certain objective functions, like the Rosenbrock function here, for which the higher order do not bound tighter than the lower order ones. If we analyze the Rosenbrock function with the artificially increased dependency problem, the second order calculation behaves as one would expect, namely requiring more steps than its higher order counter parts (see Fig. 6.7). The interval evaluation dies of the dependency problem when aiming for ๐‘  < 1E-6, which is why it is not shown in Fig. 6.7. For all calculations using QFB/LDB, the global minimum of the generalized Rosenbrock function could be bound to [โˆ’1E306, 2E-27]. This bound is very tight considering the high dimensionality and the required verified computations based on floating point numbers. The optimization variables of all calculations are contained in [0.999999998, 1.000000002] ๐‘› , which is a box of side length 4E-9 and hence almost three orders of magnitude smaller than the minimum split size. This is because QFB and LDB are not bound to splitting boxes in half, but they can decrease their size as 151 1000 QFB/LDB O2 106 QFB/LDB O3 100 QFB/LDB O4 QFB/LDB O5 10 5 10 time [s] Steps 1 104 0.1 103 0.01 2 4 3 5 6 7 2 3 4 5 6 7 dimension ๐‘› dimension ๐‘› Figure 6.7: Time consumption and number of steps in the optimization of the ๐‘› dimensional Rosenbrock function with an additional artificial dependency problem ๐‘“ = ๐‘“2D โˆ’ ๐‘“2D + ๐‘“2D for various Taylor Model orders with COSY-GO and QFB/LDB enabled. far as their rigorous methods allow them to. In summary, the example cases of the Rosenbrock functions illustrated that Taylor Model based global optimizers, and COSY-GO in particular, can handle high dimensional objective functions very efficiently. The QFB and LDB avoid the cluster effect, while the Taylor Model evaluation significantly decreases the dependency problem. For the ๐‘› = 15 dimensional non-expanded Rosenbrock function, a reduction of the search volume by a factor of more than 4E157 was accomplished in 84017 steps and less than 36 seconds (see Fig. 6.6) on an IntelยฎCoreTM i5-7200U CPU 2.5GHz. 6.2 The Lennard-Jones Potential Problem 6.2.1 Introduction In this section, the capabilities of a Taylor Model based verified global optimizer are demonstrated on the example of finding minimum energy configurations of particles when the well-known Lennard-Jones potential models their pairwise interactions. First, we introduce the Lennard-Jones potential and the principal aspects of the optimization problem. Then, we discuss the setup of the optimization problem for particle configurations in 1D before presenting the results of the verified optimization. Lastly, we discuss the more involved setup of the optimization problem in 2D and 3D 152 and present the associated verified optimization results. 6.2.1.1 The Lennard-Jones Potential ๐‘ˆLJ The 12-6 Lennard-Jones potential    ๐œŽ 12  ๐œŽ  6 ๐‘ˆLJ (๐‘Ÿ) = 4๐‘ˆ0 โˆ’ (6.6) ๐‘Ÿ ๐‘Ÿ is used as a simplified model to describe the interaction between two electrically neutral atoms or molecules with a distance ๐‘Ÿ > 0 between them. It was proposed by Lennard-Jones [50] as a specific version of the more general ๐‘Ÿ โˆ’๐‘Ž -๐‘Ÿ โˆ’๐‘ type potentials he suggested in [49] to model such interactions. The ๐‘Ÿ โˆ’12 term represents the strong repulsion of particles at very small distances. The attraction for moderate distances, which quickly decreases with larger distances, is modeled by the ๐‘Ÿ โˆ’6 term. The parameter ๐‘ˆ0 scales the depth of the potential well, which is related to the strength of the interaction between the two particles. The Van-der-Waals radius ๐œŽ is also referred to as the particle size and indicates where the sign of the potential changes. It represents the distance at which the interaction potential of the two particles assumes the same value as for the configuration where the two particles are infinitely far away from each other. โˆš6 The potential assumes its single minimum at the equilibrium distance of ๐‘Ÿ โ˜… = 2๐œŽ. For distances smaller than the equilibrium distance, the potential is strictly monotonically decreasing, and for distances larger than the equilibrium distance, the potential strictly monotonically increasing. The values ๐œŽ and ๐‘ˆ0 depend on the particles involved in the modeled pairwise interaction. For our analysis, we will only consider one sort of particle corresponding to only one set of values for ๐œŽ and ๐‘ˆ0 . To simplify the potential, we consider distances ๐‘Ÿ and ๐œŽ in units of the equilibrium distance โˆš 6 2๐œŽ, and energy in units of ๐‘ˆ0 . This yields ๐‘ˆLJ,lit = ๐‘Ÿ โˆ’12 โˆ’ 2๐‘Ÿ โˆ’6 , (6.7) a form of the Lennard-Jones potential that is often used in literature. However, we offset this potential by one for convenience of the calculations and optimization in this section. So, we define 153 the Lennard-Jones potential of two identical particles with a distance ๐‘Ÿ > 0 between them as ๐‘ˆLJ (๐‘Ÿ) = 1 + ๐‘Ÿ โˆ’12 โˆ’ 2๐‘Ÿ โˆ’6 , (6.8) โ˜… at ๐‘Ÿ โ˜… yields so that its single minimum ๐‘ˆLJ โ˜… =0 ๐‘ˆLJ and ๐‘Ÿ โ˜… = 1. (6.9) In Fig. 6.8, the single pairwise interaction potential between two identical particles from Eq. (6.8) is shown. Note the shallowness of the potential and the large range of function values. 1.4 ๐‘ˆLJ (๐‘Ÿ) 140 ๐‘ˆLJ (๐‘Ÿ) 1.2 120 potential energy ๐‘ˆLJ potential energy ๐‘ˆLJ 1 100 0.8 80 0.6 60 0.4 40 0.2 20 0 0 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 interparticle distance ๐‘Ÿ interparticle distance ๐‘Ÿ Figure 6.8: The Lennard-Jones potential for a pairwise interaction between two particles as defined in Eq. (6.8). The potential well around the minimum is shown on the left, while the right plot emphasizes its shallowness compared to the steep potential wall for ๐‘Ÿ < 1. The potential is offset for the convenience of calculation so that the single minimum ๐‘ˆLJโ˜… has an energy of zero. 6.2.1.2 Configurations of Particles S ๐‘˜ Consider a configuration S ๐‘˜ of ๐‘˜ identical particles that have their pairwise interaction modeled by the Lennard-Jones potential from Eq. (6.8). The overall interaction potential ๐‘ˆ ๐‘˜ of that configuration is given by ๐‘˜โˆ’1 โˆ‘๏ธ ๐‘˜ โˆ‘๏ธ  ๐‘ˆ๐‘˜ = ๐‘ˆLJ ๐‘Ÿ๐‘– ๐‘— , (6.10) ๐‘–=1 ๐‘—=๐‘–+1 154 the sum of all pairwise interaction potentials ๐‘ˆLJ , where ๐‘Ÿ๐‘– ๐‘— = ๐‘Ÿ ๐‘—๐‘– is the distance between the particles ๐‘๐‘– and ๐‘ ๐‘— . The number of pairwise interactions ๐‘˜ (๐‘˜ โˆ’ 1) ๐‘›pairs = (6.11) 2 roughly increases with the square of the number of particles ๐‘˜. Note that ๐‘˜โˆ’1 โˆ‘๏ธ ๐‘˜ โˆ‘๏ธ  ๐‘ˆ ๐‘˜,lit = ๐‘ˆLJ,lit ๐‘Ÿ๐‘– ๐‘— = ๐‘ˆ ๐‘˜ โˆ’ ๐‘›pairs , (6.12) ๐‘–=1 ๐‘—=๐‘–+1 allowing a direct calculation of the results in terms of ๐‘ˆ ๐‘˜,lit from our results in terms of ๐‘ˆ ๐‘˜ . We denote the global minimum of ๐‘ˆ ๐‘˜ by ๐‘ˆโ˜… ๐‘˜ . It corresponds to the lowest energy configurations S ๐‘˜โ˜…. Those minimum energy configurations are of practical importance for the formation of molecules, assuming nature is sufficiently described by this model and finds the lowest energy configurations when assembling molecules instead of just a local minimum. The minimum energy configurations and their associated minimum energy depend on whether the configurations are considered in 1D, 2D, or 3D. Thus, we will discuss those cases of minimum energy configuration in different spatial dimensionality ๐‘›dim separately. The following notation is used to distinguish between those cases of one, two, or three spatial dimensions when it is relevant. The overall interaction potential is denoted by ๐‘ˆ ๐‘˜,๐‘›dim and its global minimum by ๐‘ˆโ˜… ๐‘˜,๐‘› . The corresponding configurations are denoted by S ๐‘˜,๐‘›dim and S ๐‘˜,๐‘›โ˜… , dim dim respectively. 6.2.1.3 The Lennard-Jones Optimization Problem and its Challenges The goal of the Lennard-Jones optimization problem is the following: Given ๐‘˜ โ‰ฅ 2 identical particles with their pairwise interaction modeled by the Lennard-Jones potential from Eq. (6.8) find the global minimum of the overall interaction energy (Eq. (6.10)) and the corresponding optimal configurations. We will conduct this optimization in a verified fashion for configurations in 1D, 2D, and 3D separately. 155 This problem is particularly interesting and challenging for global optimization because the objective function is non-convex, highly nonlinear, and potentially high dimensional depending on the number of particles ๐‘˜ considered. The function values become exceedingly large when two particles get too close to each other, while the actual resulting local minima are often very shallow. This last aspect is reminiscent of the Rosenbrock function and its long and shallow valley with rapidly rising function values outside the valley. A further prominent aspect of the system is the strong interdependence โ€“ changing the position of a single particle of a ๐‘˜-particle configuration changes ๐‘˜ โˆ’ 1 interactions and their contributions to the objective function. Accordingly, the dimensionality and hence the complexity of the optimization problem can be increased as desired by simply increasing the number of particles. This interdependence also complicates the preparation of the optimization including finding appropriate variables and a tight initial search domain that is also guaranteed to contain all global solutions. Often, some regions of the search space can be excluded by showing that they cannot contain the global minimum, which decreases the initial search volume. To clearly describe our methods of choosing appropriate optimization variables and their corresponding initial search domains despite this complexity, we build them step by step and start with the analysis of 1D Lennard-Jones configurations (๐‘›dim = 1). Based on the understanding of the 1D case, we then adapt the choice of variables and their domains to describe the global optimization of configurations in 2D and 3D. Note that there will be no detailed discussion the trivial cases when ๐‘˜ โ‰ค ๐‘›dim + 1, since obvious configurations exist where every single pairwise interaction potential of the ๐‘›pairs pairwise Lennard-Jones interactions is at its minimum ๐‘ˆLJ โ˜… = 0. In other words, all distances between particles are optimal with ๐‘Ÿ๐‘– ๐‘— = ๐‘Ÿ โ˜… = 1. In particular, the configuration S2โ˜… is a line segment, ๐‘†โ˜… 3 is an equilateral triangle in 2D and 3D, and ๐‘†โ˜… 4 is a regular tetrahedron in 3D. All of these trivial configurations are of unit length with ๐‘ˆโ˜… = 0. 156 6.2.2 Minimum Energy Lennard-Jones Configurations in 1D To find minimum energy configurations of ๐‘˜ particles in 1D using verified global optimization, we first describe the solution space of all possible minimum energy configurations in terms of a set of optimization variables (see Sec. 6.2.2.1). Then, we determine special characteristics of minimum energy configurations in 1D to reduce the initial search domain. Specifically, we calculate an upper bound on the maximum distance of two adjacent particles denoted by ๐‘ฃ ๐‘ฅ,UB , and an upper bound on any ๐‘Ÿ๐‘– ๐‘— of the configuration denoted by ๐‘Ÿ ๐‘˜,UB . Further, we determine an upper bound ๐‘ˆ ๐‘˜,UB on the minimum energy, which is then used as an initial cutoff value C for the verified global optimizer (see Sec. 2.6) and for the calculation of a lower bound on ๐‘Ÿ๐‘– ๐‘— denoted ๐‘Ÿ ๐‘˜,LB . Before we present the results of the verified global optimization, we analytically evaluate the distance of adjacent particles in an infinite equidistant configuration as a reference value for the verified global optimization. 6.2.2.1 Coordinate System, Numbering Scheme, and Variable Definition in 1D Any configuration S ๐‘˜,1D of ๐‘˜ particles in 1D can be described by placing it on the ๐‘ฅ axis with the left most particle at the origin. The particles ๐‘๐‘– are numbered from 1 to ๐‘˜ according to their ๐‘ฅ position ๐‘ฅ๐‘– such that ๐‘ฅ๐‘– โ‰ค ๐‘ฅ ๐‘— for ๐‘– < ๐‘— with ๐‘ฅ 1 = 0. (6.13) Note that ๐‘ 1 is fixed to the origin and the configuration is forced to extend along the positive ๐‘ฅ axis. We denote the distance between two adjacent particles ๐‘๐‘– and ๐‘๐‘–+1 by ๐‘ฃ ๐‘ฅ,๐‘– = ๐‘ฅ๐‘–+1 โˆ’ ๐‘ฅ๐‘– โ‰ฅ 0 for ๐‘– โˆˆ {1, 2, ..., ๐‘˜ โˆ’ 1}, (6.14) and we choose ๐‘ฃ ๐‘ฅ,๐‘– as the optimization variables. The number of optimization variables for 1D configurations is denoted by ๐‘›1D,var , with ๐‘›1D,var = ๐‘˜ โˆ’ 1. (6.15) 157 In Sec. 6.2.2.7, we optimize symmetric configurations in 1D for which the number of variables, denoted by ๐‘›1D,sym,var , are roughly half of ๐‘›1D,var , with ๏ฃฑ ๏ฃด ๏ฃฒ (๐‘˜ โˆ’ 1)/2 if ๐‘˜ odd ๏ฃด ๏ฃด ๏ฃด ๐‘›1D,sym,var = , (6.16) ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๐‘˜/2 if ๐‘˜ even ๏ฃณ because for symmetric configurations ๐‘ฃ ๐‘ฅ,๐‘– = ๐‘ฃ ๐‘ฅ,๐‘˜โˆ’๐‘– . (6.17) The distance ๐‘Ÿ๐‘– ๐‘— between any two particles ๐‘๐‘– and ๐‘ ๐‘— with ๐‘– < ๐‘— can be expressed in terms of ๐‘ฃ ๐‘ฅ,๐‘– by ๐‘—โˆ’1 โˆ‘๏ธ ๐‘Ÿ ๐‘– ๐‘— = ๐‘ฅ ๐‘— โˆ’ ๐‘ฅ๐‘– = ๐‘ฃ ๐‘ฅ,๐‘› . (6.18) ๐‘›=๐‘– 6.2.2.2 Upper Bounds ๐‘Ÿ ๐‘˜,UB and ๐‘ฃ ๐‘ฅ,UB on Inter-Particle Distances of Minimum Energy Configurations in 1D Only changing the distance ๐‘ฃ ๐‘ฅ,๐‘™ between the two adjacent particles ๐‘ ๐‘™ and ๐‘ ๐‘™+1 moves the right-side subconfiguration composed of the particles ๐‘ ๐‘— with ๐‘— > ๐‘™ along the ๐‘ฅ axis, while leaving the left-side subconfiguration of particles ๐‘๐‘– with ๐‘– โ‰ค ๐‘™ unchanged. We denote all ๐‘Ÿ๐‘– ๐‘— that depend on ๐‘ฃ ๐‘ฅ,๐‘™ by ๐‘Ÿ๐‘– ๐‘— (๐‘ฃ ๐‘ฅ,๐‘™ ), which is the cases when ๐‘๐‘– belongs to the left-side and ๐‘ ๐‘— belongs to the right-side, satisfying ๐‘– โ‰ค ๐‘™ < ๐‘— (see Eq. (6.18)). If ๐‘ฃ ๐‘ฅ,๐‘™ > 1 for any ๐‘™, all ๐‘ฃ ๐‘ฅ,๐‘™ -dependent ๐‘Ÿ๐‘– ๐‘— (๐‘ฃ ๐‘ฅ,๐‘™ > 1) are at least of length ๐‘Ÿ๐‘– ๐‘— (๐‘ฃ ๐‘ฅ,๐‘™ > 1) โ‰ฅ ๐‘ฃ ๐‘ฅ,๐‘™ > 1. By setting ๐‘ฃ ๐‘ฅ,๐‘™ = 1, all ๐‘ฃ ๐‘ฅ,๐‘™ -dependent ๐‘Ÿ๐‘– ๐‘— (๐‘ฃ ๐‘ฅ,๐‘™ = 1) are shortened such that ๐‘Ÿ๐‘– ๐‘— (๐‘ฃ ๐‘ฅ,๐‘™ > 1) > ๐‘Ÿ๐‘– ๐‘— (๐‘ฃ ๐‘ฅ,๐‘™ = 1) โ‰ฅ 1, (6.19) which monotonically lowers ๐‘ˆLJ of every involved particle pair (see monotonicity argument of single Lennard-Jones potential ๐‘ˆLJ in Sec. 6.2.1.1), while leaving the uninvolved interaction energies unchanged (see Fig. 6.9). This monotonically lowers ๐‘ˆ ๐‘˜,1๐ท such that any configuration S ๐‘˜,1D with any ๐‘ฃ ๐‘ฅ,๐‘– > 1 cannot be optimal. Thus, ๐‘ฃ ๐‘ฅ,๐‘– = 1 is the upper bound on any ๐‘ฃ ๐‘ฅ,๐‘– in optimal configurations S ๐‘˜,1D โ˜… . We denote 158 ๐‘1 ๐‘2 ๐‘๐‘™ ... ๐‘ ๐‘™+1 ๐‘๐‘˜ ๐‘1 ๐‘2 ๐‘๐‘™ ...๐‘ ๐‘๐‘˜ ๐‘ฅ ๐‘™+1 ๐‘ฅ ๐‘ฃ ๐‘ฅ,๐‘™ >1 ๐‘ฃ ๐‘ฅ,๐‘™ =1 Figure 6.9: The particles ๐‘๐‘– are numbered according to their ๐‘ฅ position. The variable ๐‘ฃ ๐‘ฅ,๐‘™ denotes the distance between particle ๐‘ ๐‘™ and ๐‘ ๐‘™+1 . A configuration with any ๐‘ฃ ๐‘ฅ,๐‘™ > 1 (left picture) is never optimal, because the ๐‘ˆ ๐‘˜ can always be lowered by setting ๐‘ฃ ๐‘ฅ,๐‘™ = 1 (right picture). this upper bound with ๐‘ฃ ๐‘ฅ,UB . So, ๐‘ฃ ๐‘ฅ,UB = 1. (6.20) Additionally, this also yields an upper bound ๐‘Ÿ ๐‘˜,UB for any ๐‘Ÿ๐‘– ๐‘— within a minimum energy configuration of ๐‘˜ particles in 1D, with ๐‘˜โˆ’1 โˆ‘๏ธ ๐‘Ÿ ๐‘˜,UB = ๐‘ฃ ๐‘ฅ,UB = ๐‘˜ โˆ’ 1. (6.21) ๐‘›=1 6.2.2.3 The Upper Bound ๐‘ˆ ๐‘˜,UB on the Minimum Energy The potential ๐‘ˆ ๐‘˜,๐‘›dim of any configuration S ๐‘˜,๐‘›dim can serve as an upper bound on the minimum energy ๐‘ˆโ˜… ๐‘˜,๐‘› because by definition ๐‘ˆ ๐‘˜,๐‘›dim always satisfies dim ๐‘ˆ ๐‘˜,๐‘›dim โ‰ฅ ๐‘ˆโ˜… ๐‘˜,๐‘› . (6.22) dim We denote upper bound configurations by S ๐‘˜,๐‘›dim ,UB and their associated potential by ๐‘ˆ ๐‘˜,๐‘›dim ,UB . A good S ๐‘˜,๐‘›dim ,UB yields a tight upper bound on the minimum energy, which can then be used as an initial cutoff value C for the verified global optimizer and for the determination of a good lower bound ๐‘Ÿ ๐‘˜,LB on any ๐‘Ÿ๐‘– ๐‘— (see Sec. 6.2.2.4). Note that the argumentation so far is not specific to only 1D configurations and will be used later on for the multidimensional cases. For 1D, a good S ๐‘˜,1D,UB is given by mirroring the first half of S ๐‘˜โˆ’1,1D โ˜… onto its second half to replace it such that the resulting S ๐‘˜,1D,UB is mirror symmetric. The mirror is placed slightly off-center to generate a ๐‘˜-particle upper bound configuration from the optimal (๐‘˜ โˆ’ 1)-particle configuration. Specifically, the mirror is placed on particle ๐‘ (๐‘˜+1)/2 when ๐‘˜ is odd, and in the middle between particle ๐‘ ๐‘˜/2 and ๐‘ ๐‘˜/2+1 when ๐‘˜ is even. This mirror symmetric configuration yields an upper bound ๐‘ˆ ๐‘˜,1D,UB on ๐‘ˆโ˜… ๐‘˜,1D . 159 6.2.2.4 The Lower Bound ๐‘Ÿ ๐‘˜,LB on ๐‘Ÿ๐‘– ๐‘— Determining a lower bound ๐‘Ÿ ๐‘˜,LB on ๐‘Ÿ๐‘– ๐‘— is critical for the verified optimization. It is essential to formally show that ๐‘Ÿ๐‘– ๐‘— โ‰ฅ ๐‘Ÿ ๐‘˜,LB > 0 because the objective function is not defined for ๐‘Ÿ๐‘– ๐‘— = 0. Additionally, a lower bound often helps reducing the initial search domain of the optimization variables. To determine ๐‘Ÿ ๐‘˜,LB on ๐‘Ÿ๐‘– ๐‘— , we first determine the inverse relation between ๐‘ˆLJ and ๐‘Ÿ over the two domain sections, where the relation is bijective, namely, ๐‘Ÿ โ‰ค 1 and ๐‘Ÿ โ‰ฅ 1, denoted by ๐‘Ÿ min and ๐‘Ÿ max , respectively. Solving the quadratic equation hidden in ๐‘ˆLJ (๐‘Ÿ) from Eq. (6.8) yields ๏ฃฑ ๏ฃด โˆš โˆ’1 ๏ฃฒ ๐‘Ÿ min (๐‘ˆLJ ) = 1 + ๐‘ˆLJ 6 for 0 โ‰ค ๐‘ˆLJ ๏ฃด ๏ฃด ๏ฃด ๐‘Ÿ (๐‘ˆLJ ) = , (6.23) ๏ฃด โˆš โˆ’1 ๏ฃด ๐‘Ÿ max (๐‘ˆLJ ) = 1 โˆ’ ๐‘ˆLJ 6 for 0 โ‰ค ๐‘ˆLJ โ‰ค 1 ๏ฃด ๏ฃด ๏ฃณ where ๐‘Ÿ min โ‰ค 1 is monotonically decreasing with increasing ๐‘ˆLJ and ๐‘Ÿ max โ‰ฅ 1 is monotonically increasing with increasing ๐‘ˆLJ . Fig. 6.10 illustrates ๐‘Ÿ (๐‘ˆLJ ). 3 3 ๐‘Ÿ min (๐‘ˆLJ ) 2.5 2.5 ๐‘Ÿ max (๐‘ˆLJ ) interparticle distance ๐‘Ÿ interparticle distance ๐‘Ÿ 2 2 1.5 1.5 1 1 0.5 0.5 0 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0.01 1 100 1E4 1E6 potential energy ๐‘ˆLJ potential energy ๐‘ˆLJ Figure 6.10: The relation between ๐‘ˆLJ and the corresponding inter-particle distance(s). Note that ๐‘Ÿ max (๐‘ˆLJ ) is only defined for ๐‘ˆLJ โ‰ค 1. ๐‘Ÿ min (๐‘ˆLJ ) is only decreasing very slowly with increasing ๐‘ˆLJ as the logarithmic plot on the right shows. The potential ๐‘ˆ ๐‘˜ from any configuration S ๐‘˜ satisfies ๐‘ˆ ๐‘˜ โ‰ฅ ๐‘ˆLJ (๐‘Ÿ๐‘– ๐‘— ) (6.24) 160 for any ๐‘Ÿ๐‘– ๐‘— of S ๐‘˜ . In other words, ๐‘ˆ ๐‘˜ is an upper bound on all pairwise interactions ๐‘ˆLJ (๐‘Ÿ๐‘– ๐‘— ) of S ๐‘˜ . Having the monotonicity property of ๐‘Ÿ min , an upper bound on ๐‘ˆLJ (๐‘Ÿ๐‘– ๐‘— ) yields a lower bound on ๐‘Ÿ๐‘– ๐‘— via Eq. (6.23). Thus, ๐‘Ÿ min (๐‘ˆ ๐‘˜ ) is a lower bound on all ๐‘Ÿ๐‘– ๐‘— of S ๐‘˜ . More generally, it is a lower bound on all ๐‘Ÿ๐‘– ๐‘— in any S ๐‘˜0 for which ๐‘ˆ ๐‘˜0 โ‰ค ๐‘ˆ ๐‘˜ , specifically for ๐‘ˆ ๐‘˜0 = ๐‘ˆโ˜… ๐‘˜ . Thus, we use ๐‘Ÿ ๐‘˜,LB = ๐‘Ÿ min (๐‘ˆ ๐‘˜,UB ), (6.25) with ๐‘ˆ ๐‘˜,UB from Sec. 6.2.3.3 as a lower bound on any ๐‘Ÿ๐‘– ๐‘— in any configuration with ๐‘ˆ ๐‘˜ โ‰ค ๐‘ˆ ๐‘˜,UB , which includes all configurations considered in the optimization. Note that this method is independent of ๐‘›dim . Hence, we will also use it for configurations in 2D and 3D later on to calculate a lower bound on any ๐‘Ÿ๐‘– ๐‘— . 6.2.2.5 The Infinite 1D Equidistant Configuration Before we investigate finite minimum energy configurations of ๐‘˜ โ‰ฅ 2 particles in 1D, we derive the minimum energy state of an infinite equidistant 1D configurations [13, 52]. This one dimensional optimization problem can be solved analytically and shall serve as a reference for the results of verified optimization of finite minimum energy configurations in 1D in Sec. 6.2.2.6 and Sec. 6.2.2.7. We start our derivation by considering ๐‘˜ particles on a line, where the interaction between the particles is modeled by the Lennard-Jones potential ๐‘ˆLJ from Eq. (6.8). The distance between any two adjacent particles is a constant value ๐‘Ÿ > 0. The overall potential of such a configuration is ๐‘˜โˆ’1 โˆ‘๏ธ ๐‘ˆ ๐‘˜ (๐‘Ÿ) = (๐‘˜ โˆ’ ๐‘—) ๐‘ˆLJ ( ๐‘—๐‘Ÿ) , (6.26) ๐‘—=1 where (๐‘˜ โˆ’ ๐‘—) indicates how often an inter-particle distance of length ๐‘—๐‘Ÿ occurs in the configuration. Expanding the overall potential yields ๐‘˜โˆ’1 โˆ‘๏ธ   ๐‘ˆ ๐‘˜ (๐‘Ÿ) = (๐‘˜ โˆ’ ๐‘—) 1 + ๐‘— โˆ’12๐‘Ÿ โˆ’12 โˆ’ 2 ๐‘— โˆ’6๐‘Ÿ โˆ’6 (6.27) ๐‘—=1 ๐‘˜โˆ’1 โˆ‘๏ธ ๐‘˜โˆ’1 ๐‘˜โˆ’1 ๐‘˜โˆ’1 ๐‘˜โˆ’1 ยฉ โˆ‘๏ธ โˆ’12 โˆ‘๏ธ โˆ’11 ยช โˆ’12 ยฉ โˆ‘๏ธ โˆ’6 โˆ‘๏ธ โˆ’5 ยช โˆ’6 = (๐‘˜ โˆ’ ๐‘—) + ยญ ๐‘˜ ๐‘— โˆ’ ๐‘— ยฎ๐‘Ÿ โˆ’ 2 ยญ๐‘˜ ๐‘— โˆ’ ๐‘— ยฎ๐‘Ÿ , (6.28) ๐‘—=1 ยซ ๐‘—=1 ๐‘—=1 ยฌ ยซ ๐‘—=1 ๐‘—=1 ยฌ 161 where we denote the summation of the ๐‘— โˆ’๐‘  by the function ๐œ๐‘™ (๐‘ ) with ๐‘™ โˆ‘๏ธ ๐œ๐‘™ (๐‘ ) = ๐‘— โˆ’๐‘  = 1 + 2โˆ’๐‘  + 3โˆ’๐‘  + ... + ๐‘™ โˆ’๐‘  , for ๐‘  > 0, ๐‘™ โ‰ฅ 1. (6.29) ๐‘—=1 The function ๐œ๐‘™ (๐‘ ) satisfies ๐œ๐‘™ (๐‘ ) < ๐œ๐‘™+1 (๐‘ ) and (6.30) 1 < ๐œ๐‘™ (๐‘ 2 ) < ๐œ๐‘™ (๐‘ 1 ) for 0 < ๐‘ 1 < ๐‘ 2 and โˆ€๐‘™ > 1. (6.31) For ๐‘™ = 1, โˆ‘๏ธ1 ๐œ1 (๐‘ ) = ๐‘— โˆ’๐‘  = 1โˆ’๐‘  = 1. (6.32) ๐‘—=1 Using Eq. (6.29), we rewrite Eq. (6.28) as ๐‘˜โˆ’1 โˆ‘๏ธ ๐‘ˆ ๐‘˜ (๐‘Ÿ) = (๐‘˜ โˆ’ ๐‘—) + [๐‘˜ ๐œ ๐‘˜โˆ’1 (12) โˆ’ ๐œ ๐‘˜โˆ’1 (11)] ๐‘Ÿ โˆ’12 โˆ’ 2 [๐‘˜ ๐œ ๐‘˜โˆ’1 (6) โˆ’ ๐œ ๐‘˜โˆ’1 (5)] ๐‘Ÿ โˆ’6 . (6.33) ๐‘—=1 To find ๐‘Ÿ that minimizes ๐‘ˆ ๐‘˜ (๐‘Ÿ), we solve ๐‘‘๐‘ˆห† ๐‘˜ (๐‘Ÿ) =0 for ๐‘Ÿโ˜… > 0. (6.34) ๐‘‘๐‘Ÿ ๐‘Ÿ=๐‘Ÿโ˜…โˆˆR+ Specifically, ๐‘‘๐‘ˆห† ๐‘˜ (๐‘Ÿ) 0= = โˆ’12๐‘Ÿโ˜…โˆ’13 [๐‘˜ ๐œ ๐‘˜โˆ’1 (12) โˆ’ ๐œ ๐‘˜โˆ’1 (11)] + 12๐‘Ÿโ˜…โˆ’7 [๐‘˜ ๐œ ๐‘˜โˆ’1 (6) โˆ’ ๐œ ๐‘˜โˆ’1 (5)] ๐‘‘๐‘Ÿ ๐‘Ÿ=๐‘Ÿโ˜…โˆˆR+ 1 !1 ๐œ ๐‘˜โˆ’1 (12) โˆ’ 1๐‘˜ ๐œ ๐‘˜โˆ’1 (11) 6  ๐‘˜ ๐œ ๐‘˜โˆ’1 (12) โˆ’ ๐œ ๐‘˜โˆ’1 (11) 6 โ‡’ ๐‘Ÿโ˜… = = . (6.35) ๐‘˜ ๐œ ๐‘˜โˆ’1 (6) โˆ’ ๐œ ๐‘˜โˆ’1 (5) ๐œ ๐‘˜โˆ’1 (6) โˆ’ 1 ๐œ ๐‘˜โˆ’1 (5) ๐‘˜ As a cross-check, we evaluate ๐‘Ÿโ˜… for ๐‘˜ = 2 using Eq. (6.32) with 1 !1 ๐œ1 (12) โˆ’ 12 ๐œ1 (11) 6 1 โˆ’ 12 6 ! ๐‘Ÿโ˜… = = = 1, (6.36) ๐œ1 (6) โˆ’ 12 ๐œ1 (5) 1 โˆ’ 21 which agrees with Eq. (6.9) as expected. As a second calculation, we evaluate ๐‘Ÿโ˜… for ๐‘˜ = 3 with !1  1 1 + 2โˆ’12 โˆ’ 31 โˆ’ 13 2โˆ’11 6 2731 6 ๐‘Ÿโ˜… = = โˆˆ [0.9987241350, 0.9987241351] . (6.37) 1 + 2โˆ’6 โˆ’ 31 โˆ’ 13 2โˆ’5 2752 162 For the limit of ๐‘˜ โ†’ โˆž, we note that ๐œโˆž (๐‘ ) corresponds to the Riemann zeta function [89] โˆž โˆ‘๏ธ 1 ๐œ (๐‘ ) = , (6.38) ๐‘—๐‘  ๐‘—=1 where ๐œ‹2 ๐œ (2) = โ‰ˆ 1.644934 (6.39) 6 is known from the Basel problem [7]. With this and under consideration of Eq. (6.31), any ๐œ (๐‘ ) with ๐‘  > 2 converges to values smaller than ๐œ‹ 2 /6 but larger than 1. Specifically, ๐œ‹6 691๐œ‹ 12 ๐œ (6) = and ๐œ (12) = . (6.40) 945 638512875 Accordingly, the limit of 1 1 ๐œ ๐‘˜โˆ’1 (12) โˆ’ 1๐‘˜ ๐œ ๐‘˜โˆ’1 (11) 6 !  ๐œ (12) 6 lim ๐‘Ÿโ˜… = = ๐‘˜โ†’โˆž ๐œ ๐‘˜โˆ’1 (6) โˆ’ 1๐‘˜ ๐œ ๐‘˜โˆ’1 (5) ๐œ (6) โˆš๏ธ‚ 6 691 =๐œ‹ยท โˆˆ [0.9971792638858069273, 0.9971792638858069274] . (6.41) 675675 The upper bound ๐‘ฃ ๐‘ฅ,UB = 1 from Sec. 6.2.2.2 already told us that ๐‘Ÿโ˜… โ‰ค 1. Finding ๐‘Ÿโ˜… so close to 1 illustrates the steepness of the Lennard-Jones potential for ๐‘Ÿ < 1. 6.2.2.6 The Verified Global Optimization Results for Configurations of ๐‘˜ Particles in 1D As discussed above in Sec. 6.2.2.1, place the configuration on the positive ๐‘ฅ axis and number the particles ๐‘๐‘– from 1 to ๐‘˜ according to their ๐‘ฅ position ๐‘ฅ๐‘– such that ๐‘ฅ๐‘– โ‰ค ๐‘ฅ ๐‘— for ๐‘– < ๐‘— with ๐‘ฅ 1 = 0. (6.42) The distances ๐‘ฃ ๐‘ฅ,๐‘– = ๐‘ฅ๐‘–+1 โˆ’ ๐‘ฅ๐‘– โ‰ฅ 0 for ๐‘– โˆˆ {1, 2, ..., ๐‘˜ โˆ’ 1}, (6.43) serve as optimization variables, which yields ๐‘›1D,var = ๐‘˜ โˆ’ 1 (6.44) 163 optimization variables, as previously noted in Eq. (6.15). The distances ๐‘Ÿ๐‘– ๐‘— for the objective function are calculated from the optimization variables ๐‘ฃ ๐‘ฅ,๐‘– according to Eq. (6.18). The initial search domain of the optimization is determined by the upper and lower bound (๐‘ฃ ๐‘ฅ,UB and ๐‘ฃ ๐‘ฅ,LB ) on the distances of two adjacent particles from Sec. 6.2.2.2 and Sec. 6.2.2.4. Specifically, ๐‘ฃ ๐‘ฅ,๐‘– โˆˆ [๐‘ฃ ๐‘ฅ,LB , ๐‘ฃ ๐‘ฅ,UB ] = [๐‘Ÿ ๐‘˜,1D,LB , 1] for ๐‘– โˆˆ {1, 2, ..., ๐‘˜ โˆ’ 1}. (6.45) While the upper bound ๐‘ฃ ๐‘ฅ,UB = 1 remains unchanged for all ๐‘˜, the lower bound ๐‘ฃ ๐‘ฅ,LB = ๐‘Ÿ ๐‘˜,1D,LB depends on ๐‘ˆโ˜… ๐‘˜โˆ’1 and ๐‘ˆ ๐‘˜,UB (see Sec. 6.2.2.3 and Sec. 6.2.2.4). We start from ๐‘˜ = 2, which represents a single pair of particles. From Eq. (6.9) we know the solution of this trivial case is โ˜… = 0 and ๐‘ฃโ˜… ๐‘ˆ2,1D ๐‘ฅ,1 = 1. (6.46) We follow the method in Sec. 6.2.2.3 to construct S3,1D,UB . The mirror is placed on the particle ๐‘ 2 to mirror-copy ๐‘†โ˜… 2,1D to the right, which yields S3,1D,UB with ๐‘ฃ ๐‘ฅ,1 = ๐‘ฃ ๐‘ฅ,2 = 1. Thus, ๐‘ˆ3,1D,UB = ๐‘ˆLJ (๐‘ฃ ๐‘ฅ,1 ) + ๐‘ˆLJ (๐‘ฃ ๐‘ฅ,2 ) + ๐‘ˆLJ (๐‘ฃ ๐‘ฅ,1 + ๐‘ฃ ๐‘ฅ,3 ) (6.47) = 2๐‘ˆLJ (1) + ๐‘ˆLJ (2) = ๐‘ˆLJ (2) โ‰ค 0.968994140625. (6.48) Using this upper bound in Eq. (6.25) together with the equation for ๐‘Ÿ min in Eq. (6.23) according to the method in Sec. 6.2.2.4, we have ๐‘Ÿ 3,1D,LB = ๐‘Ÿ min (๐‘ˆ3,1D,UB ) โ‰ฅ 0.89206405909675. (6.49) Thus, the initial search domain for ๐‘˜ = 3 is ๐‘ฃ ๐‘ฅ,๐‘– โˆˆ [๐‘ฃ ๐‘ฅ,LB , ๐‘ฃ ๐‘ฅ,UB ] = [0.89206405909675, 1] for ๐‘– โˆˆ {1, 2}. (6.50) Starting with ๐‘˜ = 3, we iteratively perform the verified optimization for the ๐‘˜ particle case and use the result to calculate ๐‘ˆ ๐‘˜+1,1D,UB (see Sec. 6.2.2.3), ๐‘Ÿ ๐‘˜+1,1D,LB (see Sec. 6.2.2.4), and the initial search domain for the ๐‘˜ + 1 particle case. 164 The verified global optimization is performed with the Taylor Model based verified global optimizer COSY-GO [63, 64] in its most advanced setting with QFB/LDB enabled. An attempt to run the optimization using Interval evaluations already fails for the simplest non-trivial case of three particles in 1D. Hence, this section will only run COSY-GO in its most advanced setting. The principle algorithm of the verified global optimizer was outlined in Sec. 2.6. As a stopping condition, we use the threshold length ๐‘ min = 10โˆ’6 for all computation with COSY-GO in this chapter. A box under investigation is split into smaller boxes for further investigation unless the box is too small with all the side-lengths less than ๐‘ min . The upper bound ๐‘ˆ ๐‘˜,1D,UB from Sec. 6.2.2.3 is used as an initial cutoff value C for the optimizer. Tab. 6.1 shows the results of the verified optimization. The resulting optimized variables ๐‘ฃโ˜… ๐‘ฅ,๐‘– are listed in Tab. 6.2 and shown in Fig. 6.11. Note that the floating point inaccuracies begin to accumulate with increasing ๐‘˜ such that the bounding of ๐‘ˆโ˜… ๐‘˜,1D in Tab. 6.1 gets less and less tight. Table 6.1: Verified global optimization results for ๐‘ˆโ˜… ๐‘˜,1D . The ๐‘˜ particles form ๐‘›pairs pairwise interactions. The upper bound ๐‘ˆ ๐‘˜,1D,UB on the minimum energy (see Sec. 6.2.2.3) was used to calculate ๐‘Ÿ ๐‘˜1D,UB , which sets the lower bound of the initial search domain (see Sec. 6.2.2.4 and Eq. (6.45)). The optimizer COSY-GO with QFB/LDB enabled was operated with Taylor Models of third order. The number of remaining boxes with all side-lengths ๐‘  < ๐‘ min is denoted by ๐‘›fin . ๐‘˜ ๐‘›pairs ๐‘›fin ๐‘Ÿ ๐‘˜,1D,LB ๐‘ˆ ๐‘˜,1D,UB ๐‘ˆโ˜… ๐‘˜,1D 3 3 1 0.89206405909675 0.96899414062500 0.96887586964482 77 4 6 1 0.84674764946679 2.93492994152107 2.93486371189821 13 5 10 1 0.81433688881131 5.90034265454486 5.90034204308601589 6 15 1 0.78913201707003 9.86568839948674 9.86568807046348 29 7 21 1 0.76858739727728 14.83099005904041 14.83099004536567 39 8 28 1 0.75130633425847 20.79627461693932 20.79627460947671 30 9 36 1 0.73643514933217 27.76155137645473 27.76155137570017 69956 10 45 1 0.72341340859215 35.72682430087453 35.72682430044963 875 11 55 1 0.71185356795324 44.69209518551362 44.69209518543888 767 12 66 1 0.70147671889115 54.65736491976443 54.657364919720361862 13 78 1 0.69207564364786 65.62263397159721 65.62263397158539 307 14 91 1 0.68349232451704 77.58790260143020 77.587902601422331933 15 105 1 0.67560361208721 90.55317096078578 90.553170960781907808 165 Table 6.2: Verified global optimization results for configurations of ๐‘˜ particles in 1D. The variable ๐‘ฃโ˜… ๐‘ฅ,๐‘– is the optimal distance between two adjacent particles ๐‘๐‘– and ๐‘๐‘–+1 , with 1 โ‰ค ๐‘– < ๐‘˜, and the mirror symmetry can be observed. ๐‘˜ ๐‘– ๐‘ฃโ˜… ๐‘ฅ,๐‘– ๐‘˜ ๐‘– ๐‘ฃโ˜… ๐‘˜ ๐‘– ๐‘ฃโ˜… ๐‘˜ ๐‘– ๐‘ฃโ˜… ๐‘ฅ,๐‘– ๐‘ฅ,๐‘– ๐‘ฅ,๐‘– ๐‘˜ ๐‘– ๐‘ฃโ˜… 3 1 0.998724170 8 2 0.99729070 10 8 0.99729035 12 10 0.99729034 ๐‘ฅ,๐‘– 48 04 8990 3 2 0.998724170 14 8 0.99718036 8 3 0.99719907 885 10 9 0.99862933 01 12 11 0.99862937892 7979 14 9 0.99718124 4 1 0.99864309299 8 4 0.9971886341 11 1 0.99862934 13 1 0.99862940 066 897 889 14 10 0.99718430 4 2 0.99739647 38 8 5 0.99719907 885 11 2 0.99729033 13 2 0.99729036 373 8996 8985 14 11 0.99719782 4 3 0.99864309299 8 6 0.9972907048 11 3 0.99719782 13 3 0.99719780 25 46 30 14 12 0.99729039 5 1 0.99863238 26 8 7 0.9986294018 11 4 0.99718443 13 4 0.99718380430 8981 07 14 13 0.99862943885 5 2 0.99730683 71 9 1 0.9986293407 11 5 0.99718168 32 13 5 0.99718127076 15 1 0.99862946 5 3 0.99730683 71 9 2 0.9972904418 11 6 0.99718168 32 13 6 8047 0.99717997 881 15 2 0.99729042 5 4 0.99863238 26 9 3 0.99719822 795 11 7 0.99718443 07 13 7 0.99718047 7997 8977 15 3 0.99719785 6 1 0.99863018 03 9 4 0.9971856336 11 8 0.99719782 46 13 8 0.99718127076 20 15 4 0.99718432 6 2 0.99729422 07 9 5 0.9971856336 11 9 0.99729033 8996 13 9 0.99718430380 368 15 5 0.99718124 6 3 0.99721518 03 9 6 0.99719822 795 11 10 0.99862934897 13 10 0.99719780 30 060 9036 15 6 0.99718033 6 4 0.99729422 07 9 7 0.9972904418 12 1 0.99862937 13 11 0.99728985 7969 892 15 7 0.99718005 6 5 0.99863018 03 9 8 0.9986293407 12 2 0.99729034 13 12 0.99862940889 7941 8990 15 8 0.99718005 7941 7 1 0.99862959 10 1 0.99862933 12 3 0.99719780 14 1 0.99862943 40 01 36 433 885 9039 15 9 0.99718033 7969 7 2 0.99729148 10 2 0.99729035 12 4 0.99718389 14 2 0.99728981 30 04 15 10 0.99718124060 7 3 0.99720200 10 3 0.99719792 12 5 0.99718138 14 3 0.99719782 182 61 094 25 15 11 0.99718432368 7 4 0.99720200 10 4 0.99718474 12 6 0.99718079 14 4 0.99718373430 182 42 34 15 12 0.99719785 20 7 5 0.99729148 10 5 0.99718258 12 7 0.99718138 14 5 0.99718124 30 27 094 066 15 13 0.99729042 8977 7 6 0.99862959 10 6 0.99718474 12 8 0.99718433 14 6 0.99718036 40 42 389 7979 15 14 0.99862946881 8 1 0.99862940 10 7 0.9971979261 12 9 0.99719780 36 14 7 0.99718016 7959 18 166 ๐‘˜=3 ๐‘˜=4 -3 0.9986 ๐‘˜=5 ๐‘˜=6 0.9984 ๐‘˜=7 ๐‘˜=8 0.9982 ๐‘˜=9 -4 log10 (๐‘ฃโ˜… ๐‘ฅ,๐‘– โˆ’ ๐‘Ÿโ˜…) ๐‘˜ = 10 ๐‘˜ = 11 0.998 ๐‘˜ = 12 ๐‘ฃโ˜… ๐‘ฅ,๐‘– ๐‘˜ = 13 ๐‘˜ = 14 -5 0.9978 ๐‘˜ = 15 ๐‘Ÿโ˜… 0.9976 -6 0.9974 0.9972 -7 2 4 6 8 10 12 14 2 4 6 8 10 12 14 ๐‘– ๐‘– โ˜… Figure 6.11: The plots show the values of the optimization variables ๐‘ฃ ๐‘ฅ,๐‘– of the minimum energy configuration of ๐‘˜ particles in 1D that resulted from the verified global optimization using COSY-GO. The minimum energy configuration is mirror symmetric with the middlemost distances between adjacent particles asymptotically approaching ๐‘Ÿโ˜… โ‰ˆ 0.998724135, the solution of the infinite equidistant configuration from Eq. (6.41). The right plot shows the logarithm of the difference between the calculated distances from the verified optimization and ๐‘Ÿโ˜…. The ranges reflect the side-length of the remaining box. The left plot of Fig. 6.11 shows that the distance between adjacent particles barley changes in the middle of the configuration. However, the logarithmic plot on the right clearly shows that the distances get shorter towards the center of the configuration. The ranges in the right plot correspond to the side-length of the remaining box and its position. We observe that the optimal configurations are symmetric and the ๐‘ฃโ˜… ๐‘ฅ,๐‘– asymptotically approaches ๐‘Ÿโ˜… (Eq. (6.41)), the solution for the infinite equidistant configuration in 1D from Sec. 6.2.2.5. In Tab. 6.2, we see that the verified bounds for ๐‘˜ = 3 agree with the calculation of the equidistant configuration in Eq. (6.37). Tab. 6.3 lists the performance of COSY-GO for different Taylor Model orders. Usually, the higher the order of computation, the tighter the bounding and the lower the required number of steps, which is what we see in Tab. 6.3. At the same time, higher order computations are more time 167 demanding per step. These two factors, the computation time per step and the required number of steps, do not scale the same way with higher orders. For this particular example, calculations of order three (O3) are the most time efficient. Table 6.3: Performance of verified global optimization using COSY-GO with QFB/LDB enabled on minimum energy search of a 1D configuration of ๐‘˜ particles. The Taylor Model orders are denoted by โ€˜Oโ€™. Since QFB requires a minimum order of two, order one calculations are not listed. Computation time [s] Steps ๐‘˜ ๐‘›1D,var ๐‘›pairs O2 O3 O4 O5 O2 O3 O4 O5 3 2 3 0.011 0.017 0.010 0.015 14 10 10 8 4 3 6 0.014 0.014 0.013 0.016 24 15 14 14 5 4 10 0.027 0.021 0.021 0.016 36 23 22 18 6 5 15 0.044 0.050 0.026 0.047 57 29 28 25 7 6 21 0.085 0.043 0.034 0.079 83 34 33 33 8 7 28 0.203 0.136 0.058 0.163 130 43 42 41 9 8 36 0.560 0.136 0.101 0.332 236 60 53 52 10 9 45 1.728 0.330 0.179 0.616 475 98 66 65 11 10 55 3.373 0.747 0.438 1.828 925 176 90 81 12 11 66 7.191 0.962 0.969 3.068 1908 294 133 98 13 12 78 17.99 3.796 5.121 7.399 3975 454 292 131 14 13 91 31.96 7.276 9.480 8.928 7690 657 492 160 15 14 105 64.35 8.550 18.25 18.91 15902 994 795 199 Compared to the O2 calculation, the longer computation times of the O3 calculations per step are overcompensated by the tighter bounding and the associated reduction in the number of steps required for the verified optimization. With higher order calculations, the number of steps can be reduced even further, but the computation time per step increases significantly, such that O4 is the second most time efficient and O5 the third most time efficient despite their further reduction of calculation steps. As ๐‘˜ increases, the complexity of the problem increases quadratically as ๐‘›pairs indicates. The time efficiency of the different computation orders can change with the complexity of the objective function. 168 6.2.2.7 The Verified Global Optimization Results for Symmetric Configurations of ๐‘˜ Particles in 1D Assuming that the Lennard-Jones minimum energy configurations in 1D are indeed symmetric, this section analyzes the associated optimization problem. Considering symmetric 1D configurations roughly reduces the number of optimization variables to describe the configurations to half, since ๐‘ฃ ๐‘˜โˆ’๐‘– = ๐‘ฃ๐‘– . (6.51) This yields ๏ฃฑ ๏ฃด ๏ฃฒ (๐‘˜ โˆ’ 1)/2 if ๐‘˜ odd ๏ฃด ๏ฃด ๏ฃด ๐‘›1D,sym,var = , (6.52) ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๐‘˜/2 if ๐‘˜ even ๏ฃณ optimization variables as previously noted in Eq. (6.16). All other parameters of the optimization like the initial search domain and the method of calculated ๐‘ˆ ๐‘˜,1D,UB from Sec. 6.2.2.3 remain unchanged. As above in Sec. 6.2.2.6, we start with the trivial S2,1D โ˜… to determine ๐‘ˆ3,1D,UB (see Eq. (6.48)). We then use this upper bound and ๐‘ˆโ˜… ๐‘˜,1D in Eq. (6.25) to determine ๐‘Ÿ 3,LB,1D (see Eq. (6.49)) and with this the initial search domain (see Eq. (6.50)). Starting with ๐‘˜ = 3, we perform the verified optimization for the ๐‘˜ particle case and use the result to calculate ๐‘ˆ ๐‘˜+1,1D,UB (see Sec. 6.2.2.3), ๐‘Ÿ ๐‘˜+1,LB,1D (see Sec. 6.2.2.4), and the initial search domain for the ๐‘˜ + 1 particle case. Tab. 6.4 shows the verified results of the optimization using Taylor Models of order three. Note that the floating point inaccuracies begin to accumulate with increasing ๐‘˜ such that the bounding of ๐‘ˆโ˜…๐‘˜,1D in Tab. 6.1 gets less and less tight. For ๐‘˜ โ‰ฅ 23, the number of final boxes increases drastically. Due to the high dimensionality, ๐‘ˆ ๐‘˜ gets so shallow over the ๐‘›1D,sym,var dimensional domain that the limit of the floating point accuracy prevents narrowing down the minimum to a single final box of side-lengths ๐‘ min < 10โˆ’6 . The resulting values of the ๐‘ฃโ˜… ๐‘ฅ,๐‘– are listed in Tab. 6.5 and Tab. 6.6 below. For ๐‘˜ โ‰ฅ 23, all the resulting the final boxes are represented by one big box that contains all of them rigorously. Hence, the presented ๐‘ฃโ˜… ๐‘ฅ,๐‘– are the side-lengths of this big box. 169 Table 6.4: Verified global optimization results on the minimum energy ๐‘ˆโ˜… ๐‘˜,1D of symmetric configurations. The upper bound ๐‘ˆ ๐‘˜,1D,UB on the minimum energy (see Sec. 6.2.2.3) was used to calculate ๐‘Ÿ ๐‘˜1D,UB , which sets the lower bound of the initial search domain (see Sec. 6.2.2.4 and Eq. (6.45)). The optimizer COSY-GO with QFB/LDB enabled was operated with Taylor Models of third order. The number of remaining boxes with all side-lengths ๐‘  < ๐‘ min is denoted by ๐‘›fin . ๐‘˜ ๐‘›pairs ๐‘›fin ๐‘Ÿ ๐‘˜,1D,LB ๐‘ˆ ๐‘˜,1D,UB ๐‘ˆโ˜…๐‘˜,1D 3 3 1 0.89206405909675 0.96899414062500 0.96887586964482 77 4 6 1 0.84674764946679 2.93492994152101 2.93486371189821 13 5 10 1 0.81433688881131 5.90034265454486 5.90034204308601589 6 15 1 0.78913201707003 9.86568839948674 9.86568807046348 29 7 21 1 0.76858739727728 14.83099005904040 14.83099004536567 40 8 28 1 0.75130633425847 20.79627461693931 20.79627460947671 32 9 36 1 0.73643514933217 27.76155137645471 27.76155137570016 69958 10 45 1 0.72341340859215 35.72682430087450 35.72682430044963 878 11 55 1 0.71185356795324 44.69209518551359 44.69209518543886 772 12 66 1 0.70147671889115 54.65736491976435 54.657364919720371870 13 78 1 0.69207564364786 65.62263397159714 65.62263397158541 315 14 91 1 0.68349232451704 77.58790260143012 77.587902601422331941 15 105 1 0.67560361208721 90.55317096078569 90.553170960781907818 16 120 1 0.66831174278786 104.51843914138416 104.518439141380847582 17 136 1 0.66153786375283 119.48370720063325 119.483707200630142375 18 153 1 0.65521749055526 135.44897517554651 135.448975175543193522 19 171 1 0.64929724579321 152.41424309061438 152.414243090610400067 20 190 2 0.64373246921844 170.37951096241994 170.379510962414950320 21 210 1 0.63848543480271 189.34477880243068 189.344778802419600544 22 231 2 0.63352399921567 209.31004661870341 209.310046618696397876 23 253 2048 0.62882056259331 230.27531441704119 230.275314417024880327 24 276 4096 0.62435125909424 252.24058220167566 252.24058220160723 58147 25 300 4096 0.62009531904927 275.20584997563225 275.205849975541471139 26 325 8192 0.61603456097294 299.17111774124430 299.171117741141820645 170 Table 6.5: Verified global optimization results for symmetric configurations of ๐‘˜ particles in 1D for ๐‘˜ = 3 to ๐‘˜ = 20. ๐‘ฃโ˜… ๐‘ฅ,๐‘– is the optimal distance between two adjacent particles ๐‘๐‘– and ๐‘๐‘–+1 , and ๐‘ ๐‘˜โˆ’๐‘–โˆ’1 and ๐‘ ๐‘˜โˆ’๐‘– . The results for ๐‘˜ = 21 to ๐‘˜ = 26 are listed in Tab. 6.6. ๐‘˜ ๐‘– ๐‘ฃโ˜… ๐‘ฃโ˜… ๐‘ฃโ˜… ๐‘ฅ,๐‘– ๐‘˜ ๐‘– ๐‘ฅ,๐‘– ๐‘˜ ๐‘– ๐‘ฅ,๐‘– ๐‘˜ ๐‘– ๐‘ฃโ˜… ๐‘ฅ,๐‘– 3 1 0.998724161 ๐‘˜ ๐‘– ๐‘ฃโ˜… 10 2 0.99729031 09 13 6 0.99718040 04 16 6 0.99718024 7971 ๐‘ฅ,๐‘– 0.998643070 19 1 0.99862950 4 1 10 3 0.99719788 66 14 1 0.99862934 16 7 0.99717992 40 877 894 19 2 0.997289729046 4 2 0.99739647 38 10 4 0.99718469 14 2 9030 0.99728990 16 8 0.99717995 47 21 19 3 0.99719788 5 1 0.99863236 28 10 5 0.99718258 27 14 3 0.99719774 17 1 944 0.99862884 15 33 19 4 0.99718434 5 2 0.99730681 72 11 1 0.99862929 14 4 0.99718382422 17 2 0.99729039 361 02 8979 19 5 0.99718124 0.99863016 0.99729027 0.99718115 051 6 1 05 11 2 02 14 5 075 17 3 0.99719781 22 19 6 0.997180307957 6 2 0.99729420 09 11 3 0.99719777 51 14 6 8028 0.99717987 17 4 0.99718428 369 19 7 0.99717996 23 6 3 0.99721518 03 11 4 0.99718438 12 14 7 0.99718016 7959 17 5 0.99718119 059 19 8 0.99717982 09 7 1 0.99862956 11 5 0.99718163 37 15 1 0.99862937 17 6 8025 0.99717966 43 891 19 9 0.99717977 04 7 2 0.99729145 32 12 1 0.99862930899 15 2 0.99729032 8987 17 7 0.99717992 33 20 1 0.99862954 7 3 0.99720198 85 12 2 0.99729027 8997 15 3 0.99719775 30 17 8 0.99717981 21 873 20 2 0.997290498969 8 1 0.99862937 12 3 0.99719773 43 15 4 0.99718423377 18 1 947 0.99862880 21 427 20 3 0.99719791 11 8 2 0.99729067 12 4 0.99718396 15 5 0.99718114069 18 2 0.99729042 51 8976 20 4 0.99718438 358 8 3 0.99719888904 12 5 0.99718131 01 15 6 0.99718023 7978 18 3 0.99719784 19 20 5 0.99718128 048 8 4 0.99718863 12 6 0.99718078 35 15 7 0.99717995 50 18 4 0.99718431 365 8033 42 20 6 0.99717953 9 1 0.99862930 13 1 0.99862896932 16 1 0.99862887940 18 5 0.99718121 11 055 8027 20 7 0.99717999 19 9 2 0.99729040 13 2 0.99729028 16 2 0.99729036 18 6 0.99717962 21 8993 8983 20 8 0.99717984 05 9 3 0.99719818 13 3 0.99719773 16 3 0.99719778 18 7 0.99717993 799 37 26 28 20 9 0.99717978 899 9 4 0.99718559 13 4 0.99718423 16 4 0.99718425 18 8 0.99717980 40 387 373 15 20 10 0.99717993 881 10 1 0.99862928 13 5 0.99718119084 16 5 0.99718116064 18 9 0.99717990 898 06 171 Table 6.6: Verified global optimization results for symmetric configurations of ๐‘˜ particles in 1D for ๐‘˜ = 21 to ๐‘˜ = 26. ๐‘ฃโ˜… ๐‘ฅ,๐‘– is the optimal distance between two adjacent particles ๐‘๐‘– and ๐‘๐‘–+1 , and ๐‘ ๐‘˜โˆ’๐‘–โˆ’1 and ๐‘ ๐‘˜โˆ’๐‘– . The results for ๐‘˜ = 3 to ๐‘˜ = 20 are listed in Tab. 6.5. ๐‘˜ ๐‘– ๐‘ฃโ˜… ๐‘ฅ,๐‘– ๐‘˜ ๐‘– ๐‘ฃโ˜…๐‘ฅ,๐‘– ๐‘˜ ๐‘– ๐‘ฃโ˜…๐‘ฅ,๐‘– 21 1 0.99862958 869 22 11 0.99718002 7864 24 9 0.99717994878 21 2 0.99729053 8965 23 1 0.99862968 24 10 0.99717991874 859 21 3 0.99719795 07 23 2 9062 0.99728955 24 11 0.99717990873 21 4 441 0.99718354 23 3 0.99719805 24 12 0.99718013 7849 698 21 5 0.99718131 044 23 4 0.99718451 25 1 0.99862977 344 850 ๐‘˜ ๐‘– ๐‘ฃโ˜… 8037 ๐‘ฅ,๐‘– 21 6 0.99717949 23 5 0.99718141034 25 2 0.99729072 8946 26 5 0.99718155 21 7 0.99718002 7915 23 6 0.99718046 7939 25 3 0.99719814688 019 8060 26 6 0.99717925 21 8 0.99717988 00 23 7 0.99718011 7904 25 4 0.99718460335 26 7 0.99718025 21 9 0.99717981 894 23 8 0.99717997890 25 5 0.99718150024 7890 26 8 0.99718011 21 10 0.99717979 891 23 9 0.99717990883 25 6 8056 0.99717930 7875 26 9 0.99718004 7868 22 1 0.99862963 23 10 0.99717987880 25 7 0.99718021 7895 864 26 10 0.99718000 7864 22 2 0.99729058 23 11 0.99717985878 25 8 0.99718006 7880 8960 26 11 0.99717998863 22 3 0.997197100 24 1 0.99862973 25 9 0.99717999873 02 854 26 12 0.99717997862 22 4 446 0.99718349 24 2 0.99729067 25 10 0.99717995869 8025 8951 26 13 0.99717833 22 5 0.99718136 038 24 3 0.99719810 25 11 0.99717994868 693 22 6 8042 0.99717944 24 4 0.99718456 25 12 0.99717993867 339 22 7 0.99718007 7909 24 5 0.99718146029 26 1 0.99862982845 22 8 0.99717992 894 24 6 8051 0.99717934 26 2 9077 0.99728941 22 9 0.99717986 888 24 7 0.99718016 7899 26 3 0.99719819683 22 10 0.99717983 885 24 8 0.99718001 7884 26 4 0.99718465330 172 In Fig. 6.12, the results for the distances ๐‘ฃ ๐‘ฅ,๐‘– are shown. For ๐‘˜ > 19, the final boxes are summarized, which explains the larger ranges in the right plot. The results for symmetric 1D configurations agree with the previous results presented in Sec. 6.2.2.6, where this symmetry was not assumed. Tab. 6.7 lists the performance of COSY-GO for different Taylor Model orders, where the orders from two to five are denoted by โ€˜Oโ€™. As expected and seen already seen in Tab. 6.3, the number of required steps tends to reduce with higher order Taylor Models due to the tighter bounding capabilities. At the same time, the higher order calculations require more computation time per step, which can increase the overall computation time. For ๐‘˜ < 23, the calculations of order three (O3) are the most time efficient just like for the results in Sec. 6.2.2.6. Only for very large ๐‘˜, the verified optimization starts to struggle with the floating ๐‘Ÿโ˜… ๐‘˜ = 11 ๐‘˜ = 26 ๐‘˜ = 10 0.9986 ๐‘˜ = 25 ๐‘˜=9 0.997185 ๐‘˜ = 24 ๐‘˜=8 0.9984 ๐‘˜ = 23 ๐‘˜=7 0.997184 ๐‘˜ = 22 ๐‘˜=6 0.9982 ๐‘˜ = 21 ๐‘˜=5 ๐‘˜ = 20 ๐‘˜=4 0.997183 ๐‘˜ = 19 ๐‘˜=3 0.998 ๐‘˜ = 18 ๐‘ฃโ˜… ๐‘ฅ,๐‘– ๐‘˜ = 17 ๐‘ฃโ˜… ๐‘ฅ,๐‘– 0.997182 0.9978 ๐‘˜ = 16 ๐‘˜ = 15 0.997181 ๐‘˜ = 14 0.9976 ๐‘˜ = 13 ๐‘˜ = 12 0.997180 0.9974 0.997179 0.9972 2 4 6 8 10 12 2 4 6 8 10 12 ๐‘– ๐‘– โ˜… Figure 6.12: The plots show the values for the optimized variables ๐‘ฃ ๐‘ฅ,๐‘– of the symmetric minimum energy configuration of ๐‘˜ particles that resulted from the verified global optimization. Again, the middlemost distances asymptotically approach ๐‘Ÿโ˜… โ‰ˆ 0.998724135, the solution of the infinite equidistant configuration from Eq. (6.41). The right plot shows the logarithm of the difference between the calculated distances from the verified optimization and ๐‘Ÿโ˜…. The ranges reflect the side-length of the remaining box. 173 Table 6.7: Performance of verified global optimization using COSY-GO with QFB/LDB enabled for the minimum energy search of a 1D symmetric configuration of ๐‘˜ particles. The Taylor Model orders are denoted by โ€˜Oโ€™. Note that we use ๐‘›var as a shorthand notation for ๐‘›1D,sym,var in the table. Computation time [s] Steps ๐‘˜ ๐‘›var ๐‘›pairs O2 O3 O4 O5 O2 O3 O4 O5 3 1 3 0.010 0.010 0.016 0.011 8 7 7 5 4 2 6 0.012 0.011 0.016 0.013 17 11 10 10 5 2 10 0.013 0.011 0.016 0.013 18 12 12 10 6 3 15 0.042 0.021 0.031 0.036 28 20 19 16 7 3 21 0.032 0.029 0.016 0.025 28 20 20 18 8 4 28 0.048 0.027 0.047 0.053 40 25 25 24 9 4 36 0.045 0.025 0.047 0.065 42 25 25 24 10 5 45 0.075 0.042 0.078 0.100 59 31 29 29 11 5 55 0.099 0.058 0.094 0.365 62 33 30 29 12 6 66 0.137 0.084 0.140 0.318 95 43 35 34 13 6 78 0.277 0.088 0.187 0.474 101 46 37 34 14 7 91 0.653 0.159 0.343 0.794 171 66 45 38 15 7 105 0.473 0.173 0.406 0.906 188 66 49 39 16 8 120 1.355 0.478 0.843 1.468 313 89 70 43 17 8 136 2.043 1.163 0.641 1.746 317 94 72 45 18 9 153 3.514 1.605 1.893 2.587 573 130 104 53 19 9 171 3.515 1.983 1.991 2.433 571 130 113 59 20 10 190 8.748 3.593 4.860 4.665 1113 164 157 74 21 10 210 8.285 2.517 4.485 6.630 1101 186 170 82 22 11 231 15.47 2.635 8.952 18.40 1959 234 230 118 23 11 253 56.17 68.51 170.2 418.5 6208 4360 4340 4227 24 12 276 118.3 155.1 419.9 1255 12247 8518 8494 8399 25 12 300 130.2 175.5 467.7 1361 12503 8551 8539 8429 26 13 325 276.3 401.5 1215 3601 24527 16832 16839 15332 point accuracy and the associated increase in final boxes. The reduction of the optimization variables by assuming symmetric configurations, in comparison to the calculations in Sec. 6.2.2.6, significantly reduces the computation time and the number of steps. In Fig. 6.13, the time efficiency and the number of steps required for the optimization are shown together with the results from the previous section (Sec. 6.2.2.6). 174 O2 103 O3 O4 104 O5 O2sym 102 O3sym computation time [s] O4sym O5sym 103 101 Steps 100 102 10โˆ’1 101 10โˆ’2 10 15 5 20 25 5 10 15 20 25 number of paricles ๐‘˜ number of paricles ๐‘˜ Figure 6.13: Performance of the minimum energy search for configurations of ๐‘˜ particles in 1D using COSY-GO with different Taylor Model orders with QFB/LDB enabled. The order of the Taylor Models is denoted by โ€˜Oโ€™. The results from both Sec. 6.2.2.6 and Sec. 6.2.2.7 (โ€™symโ€™) are shown. 6.2.2.8 Redundancies and Penalty Functions Note that there are two versions for every nonsymmetric configuration in 1D that are mirror images of each other with regard to the midpoint of the configuration, i.e., for every configuration (๐‘ฃ ๐‘ฅ,1 , ๐‘ฃ ๐‘ฅ,2 , ..., ๐‘ฃ ๐‘ฅ,๐‘˜โˆ’1 ), there is a mirror configuration (๐‘ฃ ๐‘ฅ,๐‘˜โˆ’1 , ๐‘ฃ ๐‘ฅ,๐‘˜โˆ’2 , ..., ๐‘ฃ ๐‘ฅ,1 ). Both configurations are equivalent for the optimization problem, but are distinct domains in the search space of the optimizer. Because the solutions of the 1D studies are symmetric, this redundancy of mirror images of the same configuration did not appear in the results. As a preparation for the multidimensional studies below, where the solutions are not always this symmetric, we discuss a method to suppress redundant mirror images of configurations. A redundancy can be suppressed by finding criteria that distinguish those equivalent configura- tions and using a penalty function to artificially increases the objective function for the redundant 175 version(s). For mirror symmetries along the ๐‘ฅ axis, one distinction is the center of mass ๐‘˜ 1 โˆ‘๏ธ ๐‘ฅ CM = ๐‘ฅ๐‘– , (6.53) ๐‘˜ ๐‘–=1 where ๐‘ฅ๐‘– โ‰ฅ 0 and ๐‘ฅ1 = 0. Without loss of generality, one can require that the optimal configuration satisfies ๐‘ฅ ฮ”๐‘ฅ CM = ๐‘ฅ CM โˆ’ ๐‘˜ โ‰ค 0, (6.54) 2 and enforce it by letting the penalty function ๐‘ ๐‘ฅCM scale with the difference ฮ”๐‘ฅ CM if it is positive and be zero otherwise: ๏ฃฒ ๐œ† ยท ฮ”๐‘ฅCM for ฮ”๐‘ฅCM > 0 ๏ฃฑ ๏ฃด ๏ฃด ๐‘ ๐‘ฅCM = , (6.55) ๏ฃด 0 ๏ฃด otherwise ๏ฃณ where ๐œ† is a large positive number like 1010 . 6.2.3 Minimum Energy Lennard-Jones Configurations in 2D and 3D To find minimum energy configurations of ๐‘˜ particles in 2D and 3D using verified global optimization, we are going to build on the methods from the 1D studies. First, we describe the solution space of all possible minimum energy configurations in terms of a set of optimization variables (see Sec. 6.2.3.1). Then, we determine how the bounds on the minimum energy and the inter-particle distances have to be adjusted for configurations in 2D and 3D. Based on those bounds we define the initial search domain for the optimization variables and make sure that the objective function can be evaluated for every point in the initial search domain. Lastly, we present the results of the verified optimization. 6.2.3.1 Coordinate System, Numbering Scheme, and Variable Definition We use a Cartesian coordinate system (๐‘ฅ, ๐‘ฆ, ๐‘ง) to describe all possible minimum energy configurations of ๐‘˜ particles in 2D and 3D. For configurations in 2D, all ๐‘ง coordinates and variables are to be ignored. 176 The first step is placing the configuration such that the largest distance of the configuration lies on the ๐‘ฅ axis. The implications for configurations with multiple largest ๐‘Ÿ๐‘– ๐‘— are discussed below. We call the largest inter-particle distance the major axis of the configuration, denoted by ๐‘Ÿ 1๐‘˜ , because the particles that span the major axis are denoted by ๐‘ 1 and ๐‘ ๐‘˜ . The configuration is placed such that ๐‘ยฎ1 = (0, 0, 0) and ๐‘ยฎ๐‘˜ = (๐‘ฅ ๐‘˜ = ๐‘Ÿ 1๐‘˜ โ‰ฅ 0, 0, 0), (6.56) where ๐‘ยฎ๐‘– is the position vector of particle ๐‘๐‘– . This fixes ๐‘ 1 to the origin and ๐‘ ๐‘˜ to the ๐‘ฅ axis. To avoid ambiguity regarding which of the two particles of the major axis is ๐‘ 1 , we require that the center of mass of the configuration ๐‘˜ 1 โˆ‘๏ธ ๐‘ยฎCM = (๐‘ฅ CM , ๐‘ฆ CM , ๐‘งCM ) = ๐‘ยฎ๐‘– (6.57) ๐‘˜ ๐‘–=1 satisfies Eq. (6.54) for the ๐‘ฅ coordinate, i.e., ๐‘ฅ CM โˆ’ ๐‘ฅ ๐‘˜ /2 โ‰ค 0. Just like in 1D, we use the ๐‘ฅ positions of the particles to number them from 1 to ๐‘˜ such that ๐‘ฅ๐‘– โ‰ค ๐‘ฅ ๐‘— for ๐‘– < ๐‘— with ๐‘ฅ 1 = 0. (6.58) For configurations in 3D, we require that particle ๐‘ 2 is in the ๐‘ฅ๐‘ฆ plane, i.e. ๐‘ง2 = 0 (6.59) without loss of generality. To determine the orientation of the ๐‘ฆ (and ๐‘ง) axis we require without loss of generality, that ๐‘ฆ2 โ‰ค 0 and ๐‘ง3 โ‰ค 0. (6.60) The optimization variables ๐‘ฃ ๐‘ฅ,๐‘– represent the distance in the ๐‘ฅ coordinates between the particles ๐‘๐‘–+1 and ๐‘๐‘– with ๐‘ฃ ๐‘ฅ,๐‘– = ๐‘ฅ๐‘–+1 โˆ’ ๐‘ฅ๐‘– โ‰ฅ 0 for ๐‘– โˆˆ {1, 2, ..., ๐‘˜ โˆ’ 1}. (6.61) The optimization variables ๐‘ฃ ๐‘ฆ,๐‘– are the ๐‘ฆ positions of the particles with ๐‘ฃ ๐‘ฆ,๐‘– = ๐‘ฆ๐‘– for ๐‘– โˆˆ {2, 3, ..., ๐‘˜ โˆ’ 1}. (6.62) 177 For configurations in 3D, we additionally define the optimization variables ๐‘ฃ ๐‘ง,๐‘– as the ๐‘ง positions of the particles with ๐‘ฃ ๐‘ง,๐‘– = ๐‘ง๐‘– for ๐‘– โˆˆ {3, 4, ..., ๐‘˜ โˆ’ 1}. (6.63) In total, this yields ๐‘›2D,var = 2๐‘˜ โˆ’ 3 (6.64) optimization variables for configurations of ๐‘˜ particles in 2D and ๐‘›3D,var = 3๐‘˜ โˆ’ 6 (6.65) optimization variables for configurations of ๐‘˜ particles in 3D. Note that there is no variable for ๐‘ง2 because ๐‘ง2 = 0 by definition of the ๐‘ฆ axis. The squared distance ๐‘Ÿ๐‘–2๐‘— between any two particles ๐‘๐‘– and ๐‘ ๐‘— with ๐‘– < ๐‘— can be expressed in terms of ๐‘ฃ ๐‘ฅ,๐‘– , ๐‘ฃ ๐‘ฆ,๐‘– , and ๐‘ฃ ๐‘ง,๐‘– using Eq. (6.18) by ๐‘—โˆ’1 2 2 2 2 ยฉโˆ‘๏ธ ๐‘Ÿ๐‘–2๐‘— = ๐‘ฅ ๐‘— โˆ’ ๐‘ฅ๐‘– ๐‘ฃ ๐‘ฅ,๐‘› ยฎ + ๐‘ฃ ๐‘ฆ, ๐‘— โˆ’ ๐‘ฃ ๐‘ฆ,๐‘– 2 + ๐‘ฃ ๐‘ง, ๐‘— โˆ’ ๐‘ฃ ๐‘ง,๐‘– 2 . (6.66)   + ๐‘ฆ ๐‘— โˆ’ ๐‘ฆ๐‘– + ๐‘ง ๐‘— โˆ’ ๐‘ง๐‘– =ยญ ยช ยซ ๐‘›=๐‘– ยฌ ๐‘ฅ The center of mass requirement in the ๐‘ฅ coordinate ฮ”๐‘ฅ CM = ๐‘ฅ CM โˆ’ 2๐‘˜ โ‰ค 0 (from Eq. (6.54)) is enforced using the penalty function from Eq. (6.55): ๏ฃฒ ๐œ† ยท ฮ”๐‘ฅ CM for ฮ”๐‘ฅ CM > 0 ๏ฃฑ ๏ฃด ๏ฃด ๐‘ ๐‘ฅCM = (6.67) ๏ฃณ 0 otherwise ๏ฃด ๏ฃด For the largest distance requirement ฮ”๐‘Ÿ๐‘–2๐‘— = ๐‘Ÿ๐‘–2๐‘— โˆ’ ๐‘Ÿ 1๐‘˜ 2 โ‰ค 0 โˆ€๐‘– ๐‘—, (6.68) is handled using the sum of individual penalty functions for each inter-particle distance of the configuration, namely, ๐‘˜โˆ’1 ๐‘˜ ๏ฃฒ ๐œ† ยท ฮ”๐‘Ÿ๐‘–2๐‘— for ฮ”๐‘Ÿ๐‘–2๐‘— > 0 โˆ‘๏ธ โˆ‘๏ธ ๏ฃฑ ๏ฃด ๏ฃด ๐‘๐‘Ÿ 2 = ๐‘ 2 where ๐‘ 2 = . (6.69) ๐‘Ÿ ๐‘Ÿ ๐‘–๐‘— ๐‘–๐‘— 0 otherwise ๏ฃด ๏ฃด ๐‘–=1 ๐‘—=๐‘–+1 ๏ฃณ 178 The largest distance requirement reduces the rotational ambiguity in the placement of the coordinate system. Configurations with multiple largest distances often have symmetry properties that relate those largest distances to each other. If this is the case, the representations of the configuration using the coordinate system along the largest distances are identical. 6.2.3.2 Upper Bounds ๐‘Ÿ ๐‘˜,UB and ๐‘ฃ ๐‘ฅ,UB on Inter-Particle Distances of Minimum Energy Configurations in 2D and 3D The argument regarding the upper bound on the distance of two adjacent particles from Sec. 6.2.2.2 can be generalized to multidimensional cases. Using the variable and coordinate definition from above, changing a variable ๐‘ฃ ๐‘ฅ,๐‘™ moves the right-side subconfiguration composed of the particles ๐‘ ๐‘— with ๐‘— > ๐‘™ along the ๐‘ฅ axis, leaving the left-side subconfiguration of particles ๐‘๐‘– with ๐‘– โ‰ค ๐‘™ unchanged. Following the argumentation in Sec. 6.2.2.2, any configuration with any ๐‘ฃ ๐‘ฅ,๐‘™ > 1 is not optimal, thus making ๐‘ฃ ๐‘ฅ,UB = 1 (6.70) an upper bound for all ๐‘ฃ ๐‘ฅ,๐‘– in the minimum energy configuration. This shows that the result from Eq. (6.20) could be generalized to the multidimensional case. Fig. 6.14 shows this multidimensional generalization for 2D. Because the sum of all ๐‘ฃ ๐‘ฅ,๐‘– yields the length of the major axis, which is the longest distance of the configuration by definition, the upper bound ๐‘Ÿ ๐‘˜,UB = ๐‘˜ โˆ’ 1 (6.71) on any ๐‘Ÿ๐‘– ๐‘— of the minimum energy configurations is also valid for the multidimensional cases. However, for the multidimensional cases, this upper bound is not as tight as for configurations in 1D. Further advances of the method may yield tighter upper bounds on the minimum for configurations in 2D and 3D, e.g., one may be able to show that the maximum ๐‘Ÿ๐‘– ๐‘— of S ๐‘˜โ˜… in 2D can serve as ๐‘Ÿ UB of S ๐‘˜โ˜… in 3D. 179 ๐‘1 ... ๐‘๐‘˜ ๐‘1 ... ๐‘๐‘˜ ๐‘ฅ ๐‘ฅ ๐‘ฃ ๐‘ฅ,๐‘™ >1 ๐‘ฃ ๐‘ฅ,๐‘™ =1 ๐‘2 ๐‘2 ๐‘๐‘™ ๐‘๐‘™ ๐‘ ๐‘™+1 ๐‘ ๐‘™+1 Figure 6.14: The particles ๐‘๐‘– are numbered according to their ๐‘ฅ position. The variable ๐‘ฃ ๐‘ฅ,๐‘™ denotes the ๐‘ฅ distance between particle ๐‘ ๐‘™ and ๐‘ ๐‘™+1 . A configuration with any ๐‘ฃ ๐‘ฅ,๐‘™ > 1 (left picture) is never optimal, because the overall potential can always be lowered by setting ๐‘ฃ ๐‘ฅ,๐‘™ = 1 (right picture). 6.2.3.3 The Upper Bound ๐‘ˆ ๐‘˜,UB on the Minimum Energy As previously discussed in Sec. 6.2.2.3, any configuration S ๐‘˜,๐‘›dim can serve as an upper bound configuration S ๐‘˜,๐‘›dim ,UB providing an upper bound ๐‘ˆ ๐‘˜,๐‘›dim ,UB on the minimum energy ๐‘ˆโ˜… ๐‘˜,๐‘› . dim For configurations in 2D and 3D, a good upper bound configuration can be obtained by using S ๐‘˜โˆ’1,๐‘› โ˜… . We add a ๐‘˜th particle in a small and simple verified global optimization on its own, dim where only the coordinates of the ๐‘˜th particle are the optimization variables. From Sec. 6.2.3.2, we know that when we place an axis in any orientation in the minimum energy configuration, the distances between adjacent projections onto that axis are bound by 1, i.e., ๐‘ฃ ๐‘ฅ,๐‘– โ‰ค 1. Thus, the initial search domain for the โ€˜upper boundโ€™ optimization of the position of the ๐‘˜th particle is determined by the maximum and minimum coordinates of S ๐‘˜โˆ’1,๐‘› โ˜… along each orthogonal axis of the ๐‘›dim dim space, plus a band of width 1 around it (see Fig. 6.15 for a 2D example of ๐‘˜ = 6). The resulting upper bound on the overall potential of this optimized upper bound configuration S ๐‘˜,๐‘›dim ,UB is then used is then used as ๐‘ˆ ๐‘˜,๐‘›dim ,UB . 180 Figure 6.15: The optimal 2D configuration of five particles is denoted by five dots. Enclose all the five particles by a 2D rectangle using the minimum and maximum coordinates in ๐‘ฅ and ๐‘ฆ, shown by a solid line rectangle. Surround the resulting rectangle with a band of width 1, and we have a initial search domain for the sixth particle (shaded area). 6.2.3.4 Setup of Initial Search Domain of the Optimization Variables in 2D and 3D In contrast to the 1D cases, the bounds on the ๐‘Ÿ๐‘– ๐‘— do not directly translate to bounds of Cartesian search domains for configurations in 2D and 3D. As a consequence, the initial search domain covers a larger area to rigorously include all possible minimum energy configurations. The variables ๐‘ฃ ๐‘ฅ,๐‘– are bound from the top by ๐‘ฃ ๐‘ฅ,UB = 1 (Eq. (6.70)). As for the lower bound, the same argument of the 1D case is not applicable since the particle configuration is not confined on a line anymore, and we are left only with the variable definition itself, i.e., ๐‘ฃ ๐‘ฅ,๐‘– โ‰ฅ 0 (Eq. (6.61)). Thus, the initial search domain is given by   ๐‘ฃ ๐‘ฅ,๐‘– โˆˆ 0, ๐‘ฃ ๐‘ฅ,UB = [0, 1] for ๐‘– โˆˆ {1, 2, ..., ๐‘˜ โˆ’ 1}. (6.72) We bound the variables ๐‘ฃ ๐‘ฆ,๐‘– (and ๐‘ฃ ๐‘ง,๐‘– ) using the requirement that any inter-particle distance ๐‘Ÿ๐‘– ๐‘— is less than or equal to the major axis ๐‘Ÿ 1๐‘˜ and the upper bound ๐‘Ÿ UB = ๐‘˜ โˆ’ 1 (Eq. (6.71)) on any ๐‘Ÿ๐‘– ๐‘— . Consider any particle ๐‘๐‘– of the configuration with ๐‘– โˆ‰ 1, ๐‘˜. The distances ๐‘Ÿ 1๐‘– and ๐‘Ÿ๐‘–๐‘˜ must be at most ๐‘Ÿ 1๐‘˜ . Fig. 6.16 illustrates this requirement with circles of radius ๐‘Ÿ 1๐‘˜ around ๐‘ 1 and ๐‘ ๐‘˜ . 181 โˆš ๐‘Ÿ โŠฅ๐‘ฅ 3 2 ๐‘Ÿ 1๐‘˜ โˆš 3 2 ๐‘Ÿ 1๐‘˜ ๐‘1 ๐‘Ÿ 1๐‘˜ ๐‘๐‘˜ ๐‘ฅ โˆš โˆ’ 23 ๐‘Ÿ 1๐‘˜ Figure 6.16: Schematic illustration of the upper bound on distance perpendicular to the ๐‘ฅ axis (major axis) due to the requirement of having the longest distance between ๐‘ 1 and ๐‘ ๐‘˜ . ๐‘ 1 at ๐‘ฅ 1 = 0 and ๐‘ ๐‘˜ at ๐‘ฅ ๐‘˜ , where the major axis length ๐‘Ÿ 1๐‘˜ = ๐‘ฅ ๐‘˜ . All ๐‘๐‘– must be within the overlap of the two circles, which yields an upper bound on the perpendicular distance of ๐‘๐‘– to the major axis depending on ๐‘Ÿ๐‘–๐‘˜ . The upper bound corresponds to the height of an equilateral triangle of side-length ๐‘Ÿ 1๐‘˜ , which is largest for ๐‘Ÿ 1๐‘˜ = ๐‘Ÿ UB . Thus, the variables are bound by โˆš โˆš 3 3 ๐‘ฃ ๐‘ฆ,2 โˆˆ [โˆ’1, 0] ๐‘Ÿ ๐‘˜,UB and ๐‘ฃ ๐‘ฆ,๐‘– โˆˆ [โˆ’1, 1] ๐‘Ÿ ๐‘˜,UB for ๐‘– โˆˆ {3, 4, ..., ๐‘˜ โˆ’ 1} (6.73) โˆš2 โˆš 2 3 3 ๐‘ฃ ๐‘ง,3 โˆˆ [โˆ’1, 0] ๐‘Ÿ and ๐‘ฃ ๐‘ง,๐‘– โˆˆ [โˆ’1, 1] ๐‘Ÿ for ๐‘– โˆˆ {4, 5, ..., ๐‘˜ โˆ’ 1}, (6.74) 2 ๐‘˜,UB 2 ๐‘˜,UB where ๐‘Ÿ UB = ๐‘˜ โˆ’ 1 (Eq. (6.71)). Note that the initial search domains for ๐‘ฃ ๐‘ฆ,2 and ๐‘ฃ ๐‘ง3 have an upper bound of zero, because of the definition in Eq. (6.60). In Fig. 6.17, the ๐‘›2D,var -dimensional initial search domain box is shown by illustrating the initial search domain of the individual variables and how those variables relate to the position of the particles in the configuration. Note that the initial search domain does not exclude configurations with inter-particle distances ๐‘Ÿ๐‘– ๐‘— < ๐‘Ÿ ๐‘˜,LB , where ๐‘Ÿ LB is given by Eq. (6.25). In particular, it currently includes configurations 182 ๐‘ฆ โˆš 3 2 ๐‘Ÿ UB ๐‘๐‘– ๐‘1 ... ... ๐‘๐‘˜ ๐‘ฅ ๐‘3 ๐‘ ๐‘˜โˆ’1 ๐‘2 ๐‘๐‘–+1 โˆš โˆ’ 23 ๐‘Ÿ UB Figure 6.17: Initial search domain for a configuration of ๐‘˜ particles in 2D. Note that the initial domain width in ๐‘ฅ direction is always 1 (see Eq. (6.72)) and that the ๐‘ฅ position of particle ๐‘๐‘– determines the starting position in ๐‘ฅ of the domain of particle ๐‘๐‘–+1 . Particle ๐‘ 1 is fixed to the origin, particle ๐‘ 2 is bound by ๐‘ฆ 2 โ‰ค 0, and particle ๐‘ ๐‘˜ has a fixed ๐‘ฆ value of zero. with ๐‘Ÿ๐‘– ๐‘— = 0, for which the Lennard-Jones potential is not defined. To address this, we define a modified Lennard-Jones potential below without changing the optimization problem. 6.2.3.5 The Evaluation of the Objective Function We first take a closer look at how to efficiently evaluate the objective function from Eq. (6.10), which is composed of ๐‘›pairs individual Lennard-Jones interactions ๐‘ˆLJ ๐‘Ÿ๐‘– ๐‘— = 1 + ๐‘Ÿ๐‘–โˆ’12 โˆ’6  ๐‘— โˆ’ 2๐‘Ÿ๐‘– ๐‘— (6.75) as previously introduced in Eq. (6.8). Eq. (6.66) yields the squared distances ๐‘Ÿ๐‘–2๐‘— . To avoid unnecessarily taking the square-root to compute ๐‘Ÿ๐‘– ๐‘— , we implement a Lennard-Jones potential that takes the squared distance ๐‘Ÿ sqr = ๐‘Ÿ๐‘–2๐‘— as its argument with    โˆ’3 ๐‘Ÿ โˆ’3 โˆ’ 2 . ๐‘ˆLJ,sqr ๐‘Ÿ sqr = 1 + ๐‘Ÿ sqr (6.76) sqr where the squared distance ๐‘Ÿ๐‘–2๐‘— is evaluated from the optimization variables using Eq. (6.66). 183 To deal with configurations with at least one ๐‘Ÿ๐‘– ๐‘— = 0, we remind ourselves that Sec. 6.2.2.4 showed that all configurations with a single ๐‘Ÿ๐‘– ๐‘— below ๐‘Ÿ ๐‘˜,LB cannot be a minimum energy configuration. This also means that any configuration with at least one ๐‘ˆLJ (๐‘Ÿ๐‘– ๐‘— ) larger than ๐‘ˆLJ (๐‘Ÿ ๐‘˜,LB ) is not a minimum energy configuration. This leads to the idea [13] of modifying the objective function for ๐‘Ÿ๐‘– ๐‘— smaller than ๐‘Ÿ ๐‘˜,LB such that it can be evaluated for ๐‘Ÿ๐‘– ๐‘— = 0 without changing the optimization problem. The only requirement is that the modified Lennard-Jones potential ๐‘ˆหœ LJ,sqr satisfies   ๐‘ˆหœ LJ,sqr ๐‘Ÿ sqr โ‰ฅ ๐‘ˆLJ,sqr ๐‘Ÿ LB  2 2 . โˆ€๐‘Ÿ sqr < ๐‘Ÿ LB (6.77) Hence, we define the modified Lennard-Jones potential and compose it of the regular Lennard- Jones potential for ๐‘Ÿ sqr โ‰ฅ ๐‘Ÿ LB 2 and the tangential extension at ๐‘Ÿ 2 for ๐‘Ÿ 2 LB sqr โ‰ค ๐‘Ÿ LB . The modified Lennard-Jones potential [13] is then given by ๏ฃฑ 2 ๏ฃด  for ๐‘Ÿ sqr โ‰ฅ ๐‘Ÿ LB ๏ฃด ๏ฃด ๏ฃฒ ๏ฃด ๐‘ˆLJ,sqr ๐‘Ÿ sqr ๐‘ˆหœ LJ,sqr ๐‘Ÿ sqr , ๐‘Ÿ LB =        (6.78) 0 2 2 2 2 ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๐‘ˆLJ,sqr ๐‘Ÿ LB ยท ๐‘Ÿ sqr โˆ’ ๐‘Ÿ LB + ๐‘ˆLJ,sqr ๐‘Ÿ LB for ๐‘Ÿ sqr โ‰ค ๐‘Ÿ LB ๏ฃณ 0 where ๐‘ˆLJ,sqr is the first derivative of ๐‘ˆLJ,sqr with   0 ๐‘ˆLJ,sqr  โˆ’4 1 โˆ’ ๐‘Ÿ โˆ’3 . ๐‘Ÿ sqr = 6๐‘Ÿ sqr (6.79) sqr The modified Lennard-Jones potential is shown in Fig. 6.18. Next we address how to handle such a piecewise function when using Taylor Models. 6.2.3.6 Taylor Model Evaluation of Piecewise Defined Functions Consider a continuous piecewise defined function ๏ฃฒ ๐‘“L (๐‘ฅ) for ๐‘ฅ โ‰ค ๐‘ฅ 0 , ๏ฃฑ ๏ฃด ๏ฃด ๐‘“ (๐‘ฅ) = (6.80) ๏ฃด ๐‘“R (๐‘ฅ) ๏ฃด for ๐‘ฅ โ‰ฅ ๐‘ฅ 0 , ๏ฃณ with ๐‘“L (๐‘ฅ 0 ) = ๐‘“R (๐‘ฅ 0 ) , (6.81) 184 3 ๐‘ˆLJ,sqr Tangent 2.5 ๐‘ˆหœ LJ,sqr 2 potential energy 1.5 1 0.5 0 0.5 2 ๐‘Ÿ LB 1 1.5 ๐‘Ÿ sqr Figure 6.18: Piecewise defined modified Lennard-Jones potential ๐‘ˆหœ LJ,sqr shown by the black curve. The red curves shows the Lennard-Jones potential and the green line shows the tangent of this 2 = 0.92 . Lennard-Jones potential of ๐‘Ÿ sqr . The plot shown here is an example case with ๐‘Ÿ LB where ๐‘“L (๐‘ฅ) and ๐‘“R (๐‘ฅ) are ๐‘š + 1 times differentiable. We want to find a Taylor Model   ๐‘“TM = P ๐‘“ , ๐œ– ๐‘“ (6.82) that tightly captures ๐‘“ (๐‘ฅ) over the domain D = [๐‘Ž, ๐‘] where ๐‘ฅ0 โˆˆ D, and the subdomains of the function pieces are DL = [๐‘Ž, ๐‘ฅ0 ] and DR = [๐‘ฅ0 , ๐‘]. In the first step, we prepare two Taylor Models, ๐‘“L,TM and ๐‘“R,TM , for the function pieces over   the respective subdomains DL and DR . Our goal is to find ๐‘“TM = P ๐‘“ , ๐œ– ๐‘“ such that ๏ฃฑ ๏ฃฒ ๐‘“L,TM ๏ฃด ๏ฃด over DL , ๐‘“TM โŠ‡ (6.83) over ๏ฃด ๏ฃด ๐‘“R,TM ๏ฃณ DR , which is illustrated in Fig. 6.19. We start by trying to find a good polynomial P ๐‘“ that models ๐‘“ (๐‘ฅ) over the domain D well. Suppose we have polynomials that closely represent ๐‘“L over DL and ๐‘“R over DR . We denote those polynomials by PL and PR , respectively. The following weighted average of PL and PR can be a 185 ๐‘“ ๐‘“L,TM ๐‘“R,TM ๐‘“TM ๐‘Ž ๐‘ฅ0 ๐‘ ๐‘ฅ Figure 6.19: Taylor Model description of piecewise defined function. Each Taylor Model is represented by three lines as previously done in Fig. 2.2. The central curve denotes the polynomial part of the Taylor Model, while the curves above and below it indicate the bounds. good choice for P ๐‘“ , with ๐‘ค L PL + ๐‘ค R PR P๐‘“ = , (6.84) ๐‘คL + ๐‘คR where the widths of the subdomains ๐‘ค L = width (DL ) and ๐‘ค R = width (DR ) are used as weights. To perform the linear combination in Eq. (6.84), it is essential that the two polynomials, PL and PR , are based on the same expansion point and scaling, which are carried to the resulting polynomial P ๐‘“ . A natural choice is to take the midpoint ๐‘š and the half width โ„Ž = ๐‘ค/2 of the domain D as the polynomial expansion point and scaling for both polynomials. We note that the polynomial parts in ๐‘“L,TM and ๐‘“R,TM in Eq. (6.83) do not necessarily have the same expansion point and scaling discussed here. Once P ๐‘“ is found, a remainder bound ๐œ– ๐‘“ can be estimated as follows such that the requirement 186 of Eq. (6.83) is satisfied.   ๐œ– ๐‘“ = max P ๐‘“ โˆ’ ๐‘“L,TM D , P ๐‘“ โˆ’ ๐‘“R,TM D , (6.85) L R where the notation |ยท| D indicates a bound over D. As for P ๐‘“ โˆ’ ๐‘“L,TM , the expansion point and scaling of P ๐‘“ and the polynomial part of ๐‘“L,TM have to match, and the same applies to R. Typically, either P ๐‘“ or the polynomial part of ๐‘“ ๐ฟ,TM has to be adjusted to have the same expansion point and scaling. Since the latter case requires Taylor Model arithmetic for the adjustment, it would be more economical to make the necessary adjustments to P ๐‘“ . 6.2.3.7 The Verified Global Optimization Results for Configurations of ๐‘˜ Particles in 2D The coordinate system is defined for the configuration of ๐‘˜ particles in 2D according to the description in Sec. 6.2.3.1. The ๐‘ฅ axis along the major axis (the largest inter-particle distance) of the configuration is used to number the particles from 1 to ๐‘˜ according to their ๐‘ฅ position such that ๐‘ฅ๐‘– โ‰ค ๐‘ฅ ๐‘— for ๐‘– < ๐‘— . (6.86) The particle ๐‘ 1 is fixed to the origin with ๐‘ยฎ1 = (0, 0) (6.87) and particle ๐‘ ๐‘˜ is fixed to the positive ๐‘ฅ axis with ๐‘ยฎ๐‘˜ = (๐‘ฅ ๐‘˜ โ‰ฅ 0, 0). (6.88) The ๐‘ฆ axis is orientated such that ๐‘ฆ 2 โ‰ค 0. (6.89) We describe a configuration of ๐‘˜ particles in 2D using the variables ๐‘ฃ ๐‘ฅ,๐‘– = ๐‘ฅ๐‘–+1 โˆ’ ๐‘ฅ๐‘– โ‰ฅ 0 for ๐‘– โˆˆ {1, 2, ..., ๐‘˜ โˆ’ 1} and (6.90) ๐‘ฃ ๐‘ฆ,๐‘– = ๐‘ฆ๐‘– for ๐‘– โˆˆ {2, 3, ..., ๐‘˜ โˆ’ 1}, (6.91) 187 as previously defined in Sec. 6.2.3.1 in Eq. (6.61) and Eq. (6.62), respectively. This yields a total number of ๐‘›2D,var = 2๐‘˜ โˆ’ 3 (6.92) optimization variables as mentioned in Eq. (6.64). The variable domains were determined in Eq. (6.72) and Eq. (6.73) in Sec. 6.2.3.4, with ๐‘ฃ ๐‘ฅ,๐‘– โˆˆ [0, 1] for ๐‘– โˆˆ {1, 2, ..., ๐‘˜ โˆ’ 1}, (6.93) โˆš 3 ๐‘ฃ ๐‘ฆ,2 โˆˆ [โˆ’1, 0] ๐‘Ÿ ๐‘˜,UB , and (6.94) โˆš2 3 ๐‘ฃ ๐‘ฆ,๐‘– โˆˆ [โˆ’1, 1] ๐‘Ÿ for ๐‘– โˆˆ {3, 4, ..., ๐‘˜ โˆ’ 1}, with ๐‘Ÿ ๐‘˜,UB = ๐‘˜ โˆ’ 1, (6.95) 2 ๐‘˜,UB where ๐‘Ÿ ๐‘˜,UB is known from Eq. (6.71) in Sec. 6.2.3.2. As discussed in Sec. 6.2.3.5, we use the modified Lennard-Jones potential from Eq. (6.78) as the objective function without changing the optimization problem. The lower bound ๐‘Ÿ ๐‘˜,LB is determined according to Sec. 6.2.2.4. The squared inter-particle distances โ€“ the argument of this objective function โ€“ are calculated from ๐‘ฃ ๐‘ฅ,๐‘– and ๐‘ฃ ๐‘ฆ,๐‘– according to Eq. (6.66) with ๐‘ฃ ๐‘ง,๐‘– = 0. Following the center of mass requirement in the ๐‘ฅ direction and the major axis requirement from Sec. 6.2.3.1, we use the penalty functions from Eq. (6.55) and Eq. (6.69). The verified global optimization is performed with the Taylor Model based verified optimizer COSY-GO [63, 64] in its most advanced setting with QFB/LDB enabled (see Sec. 2.6). Unless stated otherwise the optimization is performed with Taylor Models of order three. The threshold length as a stopping condition is ๐‘ min = 10โˆ’6 as mentioned earlier in Sec. 6.2.2.6. We start from ๐‘˜ = 3. From Sec. 6.2.1.3, we know that the solution S3,2D โ˜… as this trivial case is an equilateral triangle, which can be represented by the particle positions โˆš ! 1 3 ๐‘ยฎ1 = (0, 0) , ๐‘ยฎ2 = , โˆ’ , ๐‘ยฎ3 = (1, 0) (6.96) 2 2 with โ˜… ๐‘ˆ3,2๐ท = 0. (6.97) 188 We follow the procedure in Sec. 6.2.3.3 to determine ๐‘ˆ4,2D,UB . For this, we optimize the position (๐‘ฅ4 , ๐‘ฆ 4 ) of a fourth particle ๐‘ 4 relative to S3,2Dโ˜… . The initial search domain for the fourth particle according to Sec. 6.2.3.3 and Fig. 6.15 is " โˆš # 3 (๐‘ฅ 4 , ๐‘ฆ 4 ) โˆˆ [โˆ’1, 2] ร— โˆ’ โˆ’ 1, 1 . (6.98) 2 The optimization yields an upper bound ๐‘ˆ4,2D,UB โ‰ค 0.9270161931701777. (6.99) Using this upper bound in Eq. (6.25) together with the equation for ๐‘Ÿ min in Eq. (6.23) according to the method in Sec. 6.2.2.4, we have  ๐‘Ÿ 4,2D,LB = ๐‘Ÿ min ๐‘ˆ4,2D,UB โ‰ฅ 0.8936896031162850. (6.100) Starting with ๐‘˜ = 4, we iteratively perform the optimization for the ๐‘˜ particle case and use the result to calculate ๐‘ˆ ๐‘˜+1,2D,UB and ๐‘Ÿ ๐‘˜+1,2D,LB for the ๐‘˜ + 1 particle case. The minimum energy configurations for four particles in 2D is shown in Fig. 6.20. The overall potential of the minimum energy configurations is bound by โ˜… ๐‘ˆ4,2D = 0.92657914153722 07. (6.101) The illustration of S4,2D โ˜… in Fig. 6.20 appears to be two connected equilateral triangles that are very slightly squisched in the horizontal direction. Tab. 6.8 lists the values for the optimal configuration S4,2D , i.e., the optimal distances between the individual particles, denoted by ๐‘Ÿ๐‘–โ˜…๐‘— , which makes the differences between S2,4D โ˜… and the structure of two connected equilateral triangles apparent. Tab. 6.8 also lists the results for the optimization variables. We can observe that S4,2D โ˜… has two symmetry axis. Compared to two equilateral triangles, S4,2D โ˜… brings the outermost particles closer together. At the same time the two particles in the middle (๐‘ 2 and ๐‘ 3 ) are slightly further apart vertically. This horizontal โ€˜squishingโ€™ of an equidistance structure to yield S4,2D โ˜… could already be observed for minimum energy configurations in 1D in Sec. 6.2.2. 189 ๐‘3 ๐‘1 ๐‘4 ๐‘2 Figure 6.20: Minimum energy configurations of four particles in 2D, S4,2D โ˜… . Interestingly, the minimum energy configuration is not a square, an obvious symmetric object with fourfold symmetry, but a rhombus with two almost equilateral triangles very slightly squished in the horizontal direction. Table 6.8: Verified global optimization results for the minimum energy configurations of four particles in 2D, S4,2D โ˜… . The ๐‘Ÿ โ˜… yield the optimal distances between particles ๐‘ and ๐‘ . ๐‘ฃโ˜… is the ๐‘–๐‘— ๐‘– ๐‘— ๐‘ฅ,๐‘– โ˜… optimal ๐‘ฅ distance between particles ๐‘๐‘– and ๐‘๐‘–+1 , and ๐‘ฃ ๐‘ฆ,๐‘– is the optimal ๐‘ฆ position of particle ๐‘๐‘– . ๐‘˜ ๐‘– ๐‘— ๐‘Ÿ๐‘–โ˜…๐‘— ๐‘˜ โ€  ๐‘– ๐‘ฃโ˜… โ€ ,๐‘– 4 1 2 0.998012409230 4 x 1 0.86312581 67 4 1 3 0.998012470230 +7 4 x 2 0.0000000โˆ’1 4 1 4 1.726251672347 4 x 3 0.86312581 67 4 2 3 1.002083071 2798 4 y 2 โˆ’0.50104139 54 4 2 4 0.998012470230 4 y 3 0.50104154 39 4 3 4 0.998012409230 The structure of the two equilateral triangles consists of five distances of 1 and one distance of โˆš โˆš๏ธ 3. Thus, its potential energy is ๐‘ˆLJ ( (3)) โ‰ˆ 0.930041152. Relative to this, all distances in S4,2D โ˜… are slightly smaller except for ๐‘Ÿ 23 . Even though a square is intuitively more symmetric than S4,2D โ˜… , it has ๐‘Ÿ and ๐‘Ÿ which are 14 23 significantly larger than 1 compared to just the large distances ๐‘Ÿ 14 as the structure of two equilateral triangles forming a rhombus. As a preparation for the optimization of ๐‘˜ = 5 particles in 2D, we use the optimal configuration 190 S4,2D โ˜… from above and the method from Sec. 6.2.3.3 to determine ๐‘ˆ5,2D,UB โ‰ค 2.822464081988782 (6.102) by optimizing the position (๐‘ฅ 5 , ๐‘ฆ 5 ) of the fifth particle relative to S4,2D โ˜… . Using this result in Eq. (6.25) together with the equation for ๐‘Ÿ min in Eq. (6.23), we have  ๐‘Ÿ 5,2D,LB = ๐‘Ÿ min ๐‘ˆ5,2D,UB โ‰ฅ 0.8484840561227015. (6.103) As a result of the verified optimization, the overall potential was bound by โ˜… ๐‘ˆ5,2D = 2.82197624549224 03. (6.104) The illustration of S5,2D โ˜… in Fig. 6.21 is indistinguishable from the formation of three equilateral triangles. Only the distances between the individual particles and the values of the optimized variables provided by Tab. 6.9 can quantify the difference to a structure of equilateral triangles. Note that the values from Tab. 6.9 confirm the existence of the vertical symmetry axis through ๐‘ 3 . ๐‘1 ๐‘3 ๐‘5 ๐‘2 ๐‘4 Figure 6.21: Minimum energy configuration of five particles in 2D, S5,2D โ˜… . The distance between the major axis particles ๐‘ 1 and ๐‘ 5 , ๐‘Ÿ 15 , is slightly below two. Particles ๐‘ 2 and ๐‘ 4 are pulled upwards, reducing their distance to each other and their distance to particles ๐‘ 1 and ๐‘ 5 . Particle ๐‘ 3 is slightly above the major axis, almost preserving the ideal distance of 1 to the particles ๐‘ 2 and ๐‘ 4 . To check that Taylor Models of order three are also the most time efficient calculation order for this Lennard-Jones optimization problem, we compare the performance of COSY-GO for different Taylor Model order for configuration of ๐‘˜ = 4 and ๐‘˜ = 5 particles in 2D in Tab. 6.10. 191 Table 6.9: Verified global optimization results for the minimum energy configurations of five particles in 2D, S5,2D โ˜… . The ๐‘Ÿ โ˜… yield the optimal distance between particles ๐‘ and ๐‘ . ๐‘ฃโ˜… is the ๐‘–๐‘— ๐‘– ๐‘— ๐‘ฅ,๐‘– โ˜… optimal ๐‘ฅ distance between particles ๐‘๐‘– and ๐‘๐‘–+1 and ๐‘ฃ ๐‘ฆ,๐‘– is the optimal ๐‘ฆ position of particle ๐‘๐‘– . ๐‘˜ ๐‘– ๐‘— ๐‘Ÿ๐‘–โ˜…๐‘— 5 1 2 0.998007235 6984 ๐‘˜ โ€  ๐‘– ๐‘ฃโ˜… โ€ ,๐‘– 5 1 3 0.996784603219 5 x 1 0.49872687 66 5 1 4 1.726795219 4649 5 x 2 0.49805409 390 5 1 5 1.993561899133 5 x 3 0.49805409 390 5 2 3 1.000010593180 5 x 4 0.49872687 66 5 2 4 0.996108166 7812 17 5 y 2 โˆ’0.86445935 5 2 5 1.726795219 4649 5 y 3 0.00269885 63 5 3 4 1.000010593180 17 5 y 4 โˆ’0.86445935 5 3 5 0.996784603219 5 4 5 0.998007235 6984 Table 6.10: Performance of verified global optimization using COSY-GO with QFB/LDB enabled on minimum energy search of a 2D configuration of ๐‘˜ particles. The Taylor Model orders are denoted by โ€˜Oโ€™. Computation time [s] Steps ๐‘˜ ๐‘›2D,var O2 O3 O4 O5 O2 O3 O4 O5 4 5 1.344 0.924 1.327 1.512 5031 3197 2935 2809 5 7 167.9 91.19 108.2 150.4 463758 295767 276720 266745 As observed in the 1D examples in Sec. 6.2.2.6 and Sec. 6.2.2.7, order three turns out to be the most time efficient calculation order. While the number of required steps is slightly improved with higher order calculations, the overall computation time is larger. For configurations of ๐‘˜ = 6 particles, the computation times on a single machine are very long. Because COSY-GO is implemented in a way that easily allows for parallel computations using MPI, we used parallel computations for the computationally time intensive cases. A critical aspect of the parallel COSY-GO is the time between processor communication and the associated load balancing. During the communication phase, the processors exchange information like redistributing their remaining domain boxes, as well as sharing their most recent cutoff values. If the time between communication is chosen to be too long, some processors will run idle without work while others 192 still have a lot of boxes to evaluate. If the time is chosen too short, too much time is wasted on communication. The timing for communication depends on multiple factors. Assume each processor runs the same repetitive code with different content and at some point, in the repetitive process, the code checks if it is time to communicate. If this is the case, the processor gathers all the information it wants to communicate and waits for all the other processors. The exchange of information happens for all involved processors. The more processors there are, the larger the communication time overhead. To evaluate a good ๐‘ก com for order three calculations, we investigate the optimization for six particles in 2D to determine S6,2Dโ˜… for various ๐‘ก com with 64 cores (2 Nodes) and 1024 cores (32 Nodes) on Cori at NERSC [72], and the performance results are shown in Tab. 6.11. Table 6.11: Performance of verified global optimization using parallel COSY-GO with QFB/LDB enabled for minimum energy search of a 2D configuration of six particles, S6,2D โ˜… . The parallel computations are run on Cori at NERSC using different communication timing ๐‘กcom . 64 cores (2 Nodes) 1024 cores (32 Nodes) ๐‘กcom [s] wall clock time [s] steps wall clock time [s] steps 1 805.17 107070040 242.49 107504313 2 754.04 107074758 215.63 107302518 4 890.24 106822057 250.40 107350076 Even though the computation power, i.e., the number of cores and nodes, was increased by a factor of 16, the computation speed increased only by a factor of 3.5. While our studies on the parallel computation performance of parallel COSY-GO are still limited, this comparison illustrates some problems associated with communication. A good scheme of load balancing is important and parallel COSY-GO takes this into account. According to our analysis, ๐‘ก com = 2 s seems efficient, so we will use it for all following parallel computations. For ๐‘˜ = 6 particles in 2D, we use S5,2D โ˜… from above and the method from Sec. 6.2.3.3 to determine ๐‘ˆ6,2D,UB โ‰ค 5.643647992876073 (6.105) 193 by optimizing the position (๐‘ฅ 6 , ๐‘ฆ 6 ) of the sixth particle relative to S5,2D โ˜… . Using this result in Eq. (6.25) together with the equation for ๐‘Ÿ min in Eq. (6.23), we have  ๐‘Ÿ 6,2D,LB = ๐‘Ÿ min ๐‘ˆ6,2D,UB โ‰ฅ 0.8164709262289850. (6.106) As a result of the verified optimization, the overall potential was bound by โ˜… ๐‘ˆ6,2D = 5.64172565099496 65. (6.107) In Fig. 6.22, S6,2D โ˜… is illustrated and Tab. 6.12 lists the distances between the particles and the associated results for the optimized variables. Note that the values from Tab. 6.12 confirm the symmetry axis through ๐‘ 2 and ๐‘ 4 . ๐‘3 ๐‘1 ๐‘4 ๐‘6 ๐‘2 ๐‘5 Figure 6.22: Minimum energy configuration of six particles in 2D, S6,2D โ˜… . The configuration shown in Fig. 6.22 is composed of four almost equilateral triangles. The connecting lines between the particle positions in Fig. 6.22 show the shape look like an envelope โˆš (upside down). In the configuration, there are nine distances close to 1, four distances close to 3 from the height of two stacked triangles, and two distances with a length of slightly less than 2. As 194 Table 6.12: Verified global optimization results for the minimum energy configurations of six particles in 2D, S6,2D โ˜… . The ๐‘Ÿ โ˜… yield the optimal distance between particles ๐‘ and ๐‘ . ๐‘ฃโ˜… is the ๐‘–๐‘— ๐‘– ๐‘— ๐‘ฅ,๐‘– โ˜… optimal ๐‘ฅ distance between particles ๐‘๐‘– and ๐‘๐‘–+1 and ๐‘ฃ ๐‘ฆ,๐‘– is the optimal ๐‘ฆ position of particle ๐‘๐‘– . ๐‘˜ โ€  ๐‘– ๐‘ฃโ˜…โ€ ,๐‘– ๐‘˜ ๐‘– ๐‘— ๐‘Ÿ๐‘–โ˜…๐‘— ๐‘˜ ๐‘– ๐‘— ๐‘Ÿ๐‘–โ˜…๐‘— 6 x 1 0.49591631 05 6 1 2 0.998217170 6865 6 2 6 1.726601007 599809 211 6 x 2 0.01435166 6 1 3 1.000179841 324 6 3 4 0.992883479 2910 6 x 3 0.48631410372 6 1 4 0.996597164 6109 6 3 5 1.989450769132 30509 6 x 4 0.49754795 73 6 1 5 1.728422004 0806 6 3 6 1.711129697 6 x 5 0.49532077 51 6 1 6 1.98945120649694 6 4 5 0.996596889384 3425 6 y 2 โˆ’0.86631644 65 6 2 3 1.726600611 205 6 4 6 0.992882964 6 y 3 0.86022432 11 6 2 4 0.995908556 7753 6 5 6 1.000179734430 6 y 4 โˆ’0.00540596622 6 2 5 0.998217526 6508 6 y 5 โˆ’0.86891684706 we already saw previously, distances larger than 1 are shorter in S6,2D โ˜… compared to a structure of actual equilateral triangles at the cost of the unit distances deviating from 1 to either smaller or larger values. The symmetry of the configuration shown in Fig. 6.22 is also captured by the values in Tab. 6.12. For the computation of S7,2D โ˜… , we use S โ˜… 6,2D from above and the method from Sec. 6.2.3.3 to determine ๐‘ˆ7,2D,UB โ‰ค 8.471671506833459 (6.108) by optimizing the position (๐‘ฅ 7 , ๐‘ฆ 7 ) of the seventh particle relative to S6,2D โ˜… . Using this result in Eq. (6.25) together with the equation for ๐‘Ÿ min in Eq. (6.23), we have  ๐‘Ÿ 7,2D,LB = ๐‘Ÿ min ๐‘ˆ7,2D,UB โ‰ฅ 0.7966957780184697. (6.109) As a result of the verified optimization, the overall potential was bound by โ˜… ๐‘ˆ7,2D = 8.46513348231309 263. (6.110) The optimization was computed in parallel on 256 cores (8 Nodes) on Cori at NERSC in 2043 seconds (wall clock time) and 852446890 steps. 195 As Fig. 6.23 illustrates, the resulting minimum energy configuration S7,2D โ˜… is highly symmetric. It is an equilateral hexagon with a side-length of about 0.996434 and an additional particle at its center. ๐‘3 ๐‘๐œ… (๐œ…, ๐œˆ) โˆˆ {(5, 6), (6, 5)} ๐‘1 ๐‘4 ๐‘7 ๐‘2 ๐‘๐œˆ Figure 6.23: Minimum energy configuration of seven particles in 2D, S7,2D โ˜… . The configuration is represented by two equivalent numbering schemes. This symmetry is further supported by the values for ๐‘Ÿ๐‘–โ˜…๐‘— in Tab. 6.13. The table also shows the results for the optimized variables. Due to the symmetry of the configuration with regard to the ๐‘ฅ axis, the optimizer finds a configuration for each of the two ambiguous numbering schemes. Specifically, particles ๐‘ 2 and ๐‘ 3 have the same ๐‘ฅ position, just like particles ๐‘ 5 and ๐‘ 6 . However, because ๐‘ฆ 2 โ‰ค 0, there is only an ambiguity in numbering the particles ๐‘ ๐œ… and ๐‘ ๐œˆ from Fig. 6.23 either with (๐œ…, ๐œˆ) = (5, 6) or (๐œ…, ๐œˆ) = (6, 5). The optimizer yields a result for each of those two numbering schemes. Since both representations are equivalent, Tab. 6.13 lists the distances and variables for (๐œ…, ๐œˆ) = (5, 6). 196 Table 6.13: Verified global optimization results for the minimum energy configurations of seven particles in 2D, S7,2D โ˜… . The ๐‘Ÿ โ˜… yield the optimal distance between particles ๐‘ and ๐‘ . ๐‘ฃโ˜… is the ๐‘–๐‘— ๐‘– ๐‘— ๐‘ฅ,๐‘– โ˜… optimal ๐‘ฅ distance between particles ๐‘๐‘– and ๐‘๐‘–+1 and ๐‘ฃ ๐‘ฆ,๐‘– is the optimal ๐‘ฆ position of particle ๐‘๐‘– . The values below are for the configuration (๐œ…, ๐œˆ) = (5, 6) from Fig. 6.23. ๐‘˜ ๐‘– ๐‘— ๐‘Ÿ๐‘–โ˜…๐‘— ๐‘˜ ๐‘– ๐‘— ๐‘Ÿ๐‘–โ˜…๐‘— ๐‘˜ โ€  ๐‘– ๐‘ฃโ˜…โ€ ,๐‘– 7 1 2 0.996434801 474 7 3 4 0.996434879 482 7 x 1 0.49821747 17 7 1 3 0.996434887 7 3 5 0.996434891 7 x 2 0.000000โˆ’01 +18 474 384 7 1 4 0.9964343695078 7 3 6 9671 1.992868964 7 x 3 0.49821745 19 7 1 5 1.725875962 023 7 3 7 1.725875962 023 7 x 4 0.49821745 19 7 1 6 1.725876111 7 4 5 0.996434879 7 x 5 0.000000โˆ’01 +18 5023 482 7 1 7 1.992870155 68738 7 4 6 0.996434879 396 7 x 6 0.49821747 17 7 2 3 1.725875631 205 7 4 7 0.996435078 4369 7 y 2 โˆ’0.86293760 82 7 2 4 0.996434879 396 7 5 6 1.725875631 205 7 y 3 0.86293782 60 7 2 5 1.992869671 7 5 7 0.996434887 7 y 4 0.000000โˆ’1 +1 8964 474 7 2 6 0.9964352354384 7 6 7 0.996434801 474 7 y 5 0.86293782 60 7 2 7 1.7258761115023 7 y 6 โˆ’0.86293760 82 6.2.3.8 The Verified Global Optimization Results for Configurations of ๐‘˜ Particles in 3D The setup for the optimization of configurations of ๐‘˜ particles in 3D requires only minor additional definitions to the setup in 2D. However, to be able to read this study without having read the previous studies, we quickly summarize the process. As discussed in Sec. 6.2.3.1, we use the center of mass of the configuration and its major axis to define the placement in the coordinate system. The ๐‘ฅ axis along the major axis is used to number the particles from 1 to ๐‘˜ according to their ๐‘ฅ position such that ๐‘ฅ๐‘– โ‰ค ๐‘ฅ ๐‘— for ๐‘– < ๐‘— . (6.111) The particle ๐‘ 1 is fixed to the origin with ๐‘ยฎ1 = (0, 0, 0) (6.112) and particle ๐‘ ๐‘˜ is fixed to the positive ๐‘ฅ axis with ๐‘ยฎ๐‘˜ = (๐‘ฅ ๐‘˜ โ‰ฅ 0, 0, 0). (6.113) 197 The ๐‘ฆ axis and the ๐‘ง axis are orientated such that ๐‘ยฎ2 = (๐‘ฅ 2 โ‰ฅ 0, ๐‘ฆ 2 โ‰ค 0, ๐‘ง2 = 0) and (6.114) ๐‘ยฎ3 = (๐‘ฅ 3 โ‰ฅ 0, ๐‘ฆ 3 , ๐‘ง3 โ‰ค 0). (6.115) We describe a configuration of ๐‘˜ particles in 3D by the variables ๐‘ฃ ๐‘ฅ,๐‘– , ๐‘ฃ ๐‘ฆ,๐‘– , and ๐‘ฃ ๐‘ง,๐‘– , with ๐‘ฃ ๐‘ฅ,๐‘– = ๐‘ฅ๐‘–+1 โˆ’ ๐‘ฅ๐‘– โ‰ฅ 0 for ๐‘– โˆˆ {1, 2, ..., ๐‘˜ โˆ’ 1}, (6.116) ๐‘ฃ ๐‘ฆ,๐‘– = ๐‘ฆ๐‘– for ๐‘– โˆˆ {2, 3, ..., ๐‘˜ โˆ’ 1}, and (6.117) ๐‘ฃ ๐‘ง,๐‘– = ๐‘ง๐‘– for ๐‘– โˆˆ {3, 4, ..., ๐‘˜ โˆ’ 1}, (6.118) as previously defined in Sec. 6.2.3.1 in Eq. (6.61), Eq. (6.62), and Eq. (6.63). This yields a total number of ๐‘›3D,var = 3๐‘˜ โˆ’ 6 (6.119) optimization variables as mentioned in Eq. (6.65). The variable domains were determined in Eq. (6.72), Eq. (6.73), and Eq. (6.74) in Sec. 6.2.3.4, with ๐‘ฃ ๐‘ฅ,๐‘– โˆˆ [0, 1] for ๐‘– โˆˆ {1, 2, ..., ๐‘˜ โˆ’ 1}, (6.120) โˆš 3 ๐‘ฃ ๐‘ฆ,2 , ๐‘ฃ ๐‘ง,3 โˆˆ [โˆ’1, 0] ๐‘Ÿ ๐‘˜,UB , (6.121) โˆš2 3 ๐‘ฃ ๐‘ฆ,๐‘– โˆˆ [โˆ’1, 1] ๐‘Ÿ UB for ๐‘– โˆˆ {3, ..., ๐‘˜ โˆ’ 1}, and (6.122) โˆš2 3 ๐‘ฃ ๐‘ง,๐‘– โˆˆ [โˆ’1, 1] ๐‘Ÿ for ๐‘– โˆˆ {4, ..., ๐‘˜ โˆ’ 1} with ๐‘Ÿ UB = ๐‘˜ โˆ’ 1, (6.123) 2 UB where ๐‘Ÿ UB is known from Eq. (6.71) in Sec. 6.2.3.2. As discussed in Sec. 6.2.3.5, we use the modified Lennard-Jones potential from Eq. (6.78) as the objective function without changing the optimization problem. The lower bound ๐‘Ÿ ๐‘˜,LB is determined according to Sec. 6.2.2.4. The squared inter-particle distances โ€“ the argument of this objective function โ€“ are calculated from ๐‘ฃ ๐‘ฅ,๐‘– , ๐‘ฃ ๐‘ฆ,๐‘– , and ๐‘ฃ ๐‘ง,๐‘– according to Eq. (6.66). Following the center of mass requirement in the ๐‘ฅ direction and the major axis requirement from Sec. 6.2.3.1, we use the penalty functions from Eq. (6.55) and Eq. (6.69). 198 The verified global optimization is performed with the Taylor Model based verified optimizer COSY-GO [63, 64] in its most advanced setting with QFB/LDB enabled (see Sec. 2.6). Unless stated otherwise the optimization is performed with Taylor Models of order three. The threshold length as a stopping condition is ๐‘ min = 10โˆ’6 as mentioned earlier in Sec. 6.2.2.6. We start from ๐‘˜ = 4. From Sec. 6.2.1.3, we know that the solution S4,3D โ˜… as this trivial case is a regular tetrahedron. We note that in literature, the optimization of four particles in 3D is often used as a toy problem, which we discuss in the appendix A.1. For the computation of ๐‘ˆ5,3D,UB , we follow the procedure in Sec. 6.2.3.3 and represent S4,3D โ˜… by the particle positions โˆš ! โˆš โˆš๏ธ‚ ! 1 3 1 3 2 ๐‘ยฎ1 = (0, 0, 0) , ๐‘ยฎ2 = , โˆ’ , 0 , ๐‘ยฎ3 = , โˆ’ , โˆ’ , and ๐‘ยฎ4 = (1, 0, 0) (6.124) 2 2 2 6 3 with โ˜… ๐‘ˆ4,3D = 0. (6.125) Then, we optimize the position (๐‘ฅ 5 , ๐‘ฆ 5 ) of a fifth particle ๐‘ 5 relative to S4,3D โ˜… . The initial search domain for ๐‘ 5 according to Sec. 6.2.3.3 and Fig. 6.15 is " โˆš # " โˆš๏ธ‚ # 3 2 ๐‘ยฎ5 = (๐‘ฅ 5 , ๐‘ฆ 5 , ๐‘ง5 ) โˆˆ [โˆ’1, 2] ร— โˆ’1 โˆ’ , 1 ร— โˆ’1 โˆ’ ,1 . (6.126) 2 3 The optimization yields an upper bound ๐‘ˆ5,3D,UB โ‰ค 0.8968457347060826 (6.127) Using this upper bound in Eq. (6.25) together with the equation for ๐‘Ÿ min in Eq. (6.23) according to the method in Sec. 6.2.2.4, we have  ๐‘Ÿ 5,3D,LB = ๐‘Ÿ min ๐‘ˆ5,3D,UB โ‰ฅ 0.8948940496635427. (6.128) As a result of the verified optimization, the overall potential was bound by โ˜… ๐‘ˆ5,3D = 0.89614758429245 18. (6.129) 199 The optimization was computed in parallel on 64 cores (2 Nodes) on Cori at NERSC in 74.28 seconds (wall clock time) and 2466118 steps. As Fig. 6.24 illustrates, S5,3D โ˜… is very similar to a regular double-tetrahedron. This is further supported by the values of the optimization variables and ๐‘Ÿ๐‘–โ˜…๐‘— in Tab. 6.14. xy xz ๐‘4 yz ๐‘4 ๐‘ 3 /๐‘ 4 ๐‘1 ๐‘5 ๐‘1 ๐‘2 ๐‘5 ๐‘2 ๐‘ 1 /๐‘ 5 ๐‘3 ๐‘3 ๐‘2 Figure 6.24: Minimum energy configuration of five particles in 3D, S5,3D โ˜… . The configuration is shown in 2D projections, the ๐‘ฅ๐‘ฆ plane projection (left), the ๐‘ฅ๐‘ง plane projection (middle), and the ๐‘ฆ๐‘ง plane projection (right). The solution consists of a central equilateral triangle spanned by the particles ๐‘ 2 , ๐‘ 3 , and ๐‘ 4 in the ๐‘ฆ๐‘ง plane, and one particle each centered above and below that triangle. In other words, it is similar to a double tetrahedron, which is slightly squished along the major axis (the ๐‘ฅ axis) increasing the side-length of the equilateral triangle in the middle to values slightly larger than one. The inter-particle distances are shown in Tab. 6.14. Table 6.14: Verified global optimization results for the minimum energy configurations of five particles in 3D, S5,3D โ˜… . The ๐‘Ÿ โ˜… yield the optimal distance between particles ๐‘ and ๐‘ . ๐‘ฃโ˜… is the ๐‘–๐‘— ๐‘– ๐‘— ๐‘ฅ,๐‘– โ˜… โ˜… optimal ๐‘ฅ distance between particles ๐‘๐‘– and ๐‘๐‘–+1 . ๐‘ฃ ๐‘ฆ,๐‘– and ๐‘ฃ ๐‘ง,๐‘– are the optimal ๐‘ฆ and ๐‘ง positions of particle ๐‘๐‘– . ๐‘˜ ๐‘– ๐‘— ๐‘Ÿ๐‘–โ˜…๐‘— ๐‘˜ โ€  ๐‘– ๐‘ฃโ˜… โ€ ,๐‘– 5 1 2 0.9979070076764 5 x 1 0.8133358769 5 1 3 0.9979071596694 +11 5 x 2 0.000000โˆ’01 5 1 4 0.9979072426694 +11 5 x 3 0.000000โˆ’01 5 1 5 1.626671944 396 5 x 4 0.8133358769 5 2 3 1.001453817 231 5 y 2 โˆ’0.57818937 55 5 2 4 1.001453817 231 5 y 3 0.2890949056 5 2 5 0.9979071726764 5 y 4 0.2890949056 5 3 4 1.001453816 232 61 5 z 3 โˆ’0.50072691 5 3 5 0.9979071596694 5 z 4 0.5007269161 5 4 5 0.9979070776694 200 Even though particles ๐‘ 2 , ๐‘ 3 and ๐‘ 4 all seem to have the same ๐‘ฅ coordinate, there is no ambiguous numbering scheme due to the definition of the coordinate system with ๐‘ฆ 2 โ‰ค 0, ๐‘ง2 = 0 and ๐‘ง3 โ‰ค 0. The regular double tetrahedron consists of nine unit distances and the major axis of length โˆš๏ธ 2 2/3. S5,3D โ˜… is a slightly โ€˜squishedโ€™ version of the regular double tetrahedron along the major axis. The major axis and the distances in the direction of the major axis are shortened. Only the three inter-particle distances between ๐‘ 2 , ๐‘ 3 , and ๐‘ 4 are slightly longer than unit length and form an equilateral triangle. 6.2.4 Summary This section illustrated the many critical aspects of verified global optimization and the capabilities of COSY-GO in its most advanced setting QFB/LDB. Despite the high dimensionality, the strong interdependence, and nonlinearity of the optimization problem, COSY-GO was able to rigorously determine the minimum energy configurations. In Tab. 6.15, we summarize the results for the global minimum of Lennard-Jones configurations in 2D and 3D and also provide the corresponding values for ๐‘ˆ ๐‘˜,lit,๐‘›โ˜… using Eq. (6.12) for easier dim comparison with literature. Table 6.15: Summary of verified global optimization results on the minimum energy of configurations in 2D and 3D. ๐‘›dim ๐‘˜ ๐‘›pairs ๐‘ˆโ˜…๐‘˜,๐‘› ๐‘ˆโ˜…๐‘˜,lit,๐‘› = ๐‘ˆโ˜…๐‘˜,๐‘› โˆ’ ๐‘›pairs dim dim dim 2 4 6 0.92657914153722 07 โˆ’5.07342085846278 93 2 5 10 2.82197624549224 03 โˆ’7.17802375450776 97 2 6 15 5.64172565099496 65 โˆ’9.35827434900504 35 2 7 21 8.46513348231309 263 โˆ’12.53486651768691 737 3 5 10 0.89614758429245 18 โˆ’9.10385241570755 82 201 6.3 Verified Stability Analysis of Dynamical Systems Verified calculations are particularly important for the stability analysis of dynamical systems. With a verified upper bound on the rate of divergence, a systemโ€™s long term stability can be rigorously estimated. Both of the previously discussed applications in Chapter 4 and Chapter 5 will benefit to different degrees from such verified stability estimates. 6.3.1 The Potential Implications for the Bounded Motion Problem For the bounded motion orbits under zonal perturbation in the Earthโ€™s gravitational field (see Chapter 4), a stability estimate is the maximum rate at which two bounded orbits drift apart. Below we want to list aspects to consider for the calculation of such a verified upper bound on the rate of divergence. The bounded motion conditions from Sec. 4.2.5 require that the average nodal period ๐‘‡ ๐‘‘ and the average drift of the ascending node ฮ”ฮฉ of two bounded orbits are the same. In other words, two orbits drift apart if those two averaged quantities are not the same for the two orbits. Additionally, each of the orbits might be diverging on its own by slowly increasing or decreasing its distance from the Earth. A verified upper bound on each of those diverging factors must be determined to combine them to an overall verified upper bound on the rate at which the two bounded orbits drift apart. An upper bound on the radial drift rate of the bounded orbits moving apart is determined by the maximum difference between the individual radial drifts of each of the bounded orbits. The normal form defect of the radial phase space can be used as a measure for this radial drift. However, both the maximum and the minimum normal form defect of each orbit are relevant to determine the worst-case scenario of one of the orbits decreasing its amplitude and one of the orbits increasing its amplitude. The longitudinal drift rate of the bounded orbits moving apart is determined by the difference in the average revolution frequency of the orbital planes around the symmetry axis. The revolution frequency is proportional to the drift of the ascending node ฮ”ฮฉ per nodal period ๐‘‡ ๐‘‘ . Since both of 202 these quantities are oscillating at the same rate, the average revolution frequency can be calculated as the ratio of the average drift of the ascending node ฮ”ฮฉ and the average nodal period ๐‘‡ ๐‘‘ . Even if the orbital planes of the two bounded orbits are not radially or longitudinally drifting apart, the satellites on those orbits might still be drifting apart due to different average nodal periods, which constitutes the third drift factor. These three factors have to be taken into account and rigorously estimated to calculate an overall maximum drift rate. The combination of the individual factors is not trivial since they are not independent of each other, e.g., the individual radial drifts of the orbits have nonlinear influences on the bounded motion quantities ฮ”ฮฉ and ๐‘‡ ๐‘‘ . Verified global optimization of the overall drift rate is required to determine the maximum rate of divergence for any possible combination of the individual radial drift rates. Given that the overall maximum drift rate is formally defined, we need to determine verified versions of the involved quantities. Accordingly, the starting point of the rigorous calculation of the maximum drift rate is a rigorous map of the system. The map is based on the equations of motion of the system, which include the zonal coefficients of the Earthโ€™s gravitational potential based on measurements. To be rigorous it has to be decided if these coefficients are assumed to be exact or if the uncertainty about these coefficients is considered in the calculation. Given that the approach from Chapter 4 considers the zonal problem, ignoring sectional and tesseral terms, it seems reasonable to consider an idealized system where these coefficients are assumed to be exact. In the next step, the verified integration of the equations of motion is required to calculate a verified map representation of the system [22, 28]. In our approach (see Chapter 4), we express the vertical momentum component ๐‘ฃ ๐‘ง in terms of the other variables and system parameters. This operation includes the calculation of an inverse, which requires special methods to be performed rigorously [41, 20]. For the projection of the transfer map onto the Poincarรฉ surface representing a generalized ascending node state, another rigorous computation of an inverse is required [41, 20]. Additionally, every step of the normal form based averaging procedure from Sec. 4.3.4 for the 203 determination of the averaged quantities ฮ”ฮฉ and ๐‘‡ ๐‘‘ has to be performed rigorously. The approach then calls for another inversion to calculate the constants of motion H๐‘ง and ๐ธ as a function of the phase space variables such that the averaged bounded motion quantities match between any two orbits in the phase space. If all those procedures are performed rigorously, one can calculate rigorous bounds on the normal form defect of the system, which can then be used together with the rigorous estimations of the averaged quantities ฮ”ฮฉ and ๐‘‡ ๐‘‘ to calculate the rigorous overall rate of divergences. In summary, much effort is required to establish a verified upper bound on the maximum rate at which bounded orbits of the zonal problem drift apart. However, the practical implications of such an estimate are limited since the approach does not consider the fully perturbed system. Accordingly, we want to focus our attention on the application of a rigorous stability analysis for the system discussed in Chapter 5. 6.3.2 The Implications for the Stability Analysis of the Muon ๐‘”-2 Storage Ring A verified stability estimate of the Muon ๐‘”-2 Storage Ring can be obtained from the verified maximum rate at which particles escape the storage region of the storage ring. A measure of this rate of divergence is the normal form defect. In Chapter 5, we saw that the size of the normal form defect that a particle encounters correlates with its likelihood of getting lost. As mentioned before and discussed in [85], the number of lost particles is very important for this high precision experiment, because the losses introduce a systematic bias for the average polarization of the remaining particles, which influences the overall result of the measurement. Below we want to analyze the aspects to consider for the calculation of such a verified upper bound on the rate of divergence in form of the normal form defect using Taylor Model based verified global optimization. For the fully verified normal form defect analysis, we require a verified phase space map of the storage ring. As already mentioned before, there are many intricacies to consider for a fully rigorous map calculation. A major challenge regarding the verified calculation of the storage ring map is 204 the verified representation of every storage ring component, including all its perturbations, e.g., perturbations from ESQ fringe fields and imperfection in the magnetic field. Because further work is required to generate such a fully verified map of the Muon ๐‘”-2 Storage Ring, we will proceed with the nonverified tenth order map from Chapter 5 with an ESQ voltage of 18.3 kV. Assuming this map captures all of the relevant dynamics, the difference between using a verified map and a nonverified map is very small. To assess whether our computation order is high enough to capture the relevant dynamics, we estimate inaccuracies in the map by computing maps of various orders and showing that these inaccuracies โ€“ the main numerical error which is not based on measurement errors โ€“ are sufficiently small and will not affect the analysis result in a meaningful way when using the storage ring map of order ten. For comparison, we will additionally analyze a storage ring map that considers an ESQ voltage of 17.5 kV instead of 18.3 kV. The tunes of particles under the influence of an ESQ voltage of 17.5 kV are further away from the vertical 1/3 resonance tune. Accordingly, we expect no period-3 fixed point structures with their unstable fixed points and therefore less diverging behavior for this map compared to the 18.3 kV map. The goal is to rigorously analyze the stability of the entire five dimensional storage phase space (๐‘ฅ, ๐‘Ž, ๐‘ฆ, ๐‘, ๐›ฟ ๐‘) of the storage ring maps using verified global optimization of the normal form defect. In Sec. 6.3.3, we specify the normal form defect function as the objective function of the optimization problem. To be able to distinguish the diverging behavior in different areas of the storage region, we divide the five dimensional space into partitions. Each of those partitions is then used as the search domain for the verified global optimizer to find the maximum normal form defect in it. In Sec. 6.3.4, we present the onion layer approach [29, 13], which divides the storage region according to the dynamics in the phase space. Next, we illustrate the complexity and strong nonlinearity of the normal form defect in multiple such onion layers and how it changes for different phase space regions and ESQ voltages (see Sec. 6.3.5). In Sec. 6.3.6, the results of the verified global optimization for the two maps with the different voltages are presented and compared to each other and the results of a nonverified analysis. 205 6.3.3 The Normal Form Defect as the Objective Function for the Optimization In Sec. 2.4, the normal form defect for the propagation of a state ๐‘งยฎ with a map M was introduced as the difference between the normal form radius of the mapped state M (ยฎ๐‘ง) and the normal form radius of the original state ๐‘งยฎ. If the motion occurs in multiple phase space dimensions, there is some ambiguity to the term โ€˜normal form radiusโ€™ and the associated normal form defect. From the definition and algorithms of normal form transformations discussed in Sec. 2.3 and Sec. 2.4, it follows that there is a normal form radius for each normal form phase space. Each of these radii, yields the radius of the circular motion in this particular normal form phase space with โˆš๏ธƒ ๐‘ž NF,i (ยฎ๐‘ง0 ) 2 + ๐‘ NF,i (ยฎ๐‘ง0 ) 2 .   ๐‘Ÿ NF,๐‘– (ยฎ๐‘ง0 ) = (6.130) Accordingly, as defined in Sec. 2.4, there is a normal form defect defined for each of those normal form radii, with ๐‘‘NF,๐‘– (ยฎ๐‘ง 0 ) = ๐‘Ÿ NF,๐‘– ( M (ยฎ๐‘ง0 )) โˆ’ ๐‘Ÿ NF,๐‘– (ยฎ๐‘ง0 ) . (6.131) Additionally, we define the (overall) normal form radius of the motion as the Euclidean distance โˆš๏ธ„โˆ‘๏ธ ๐‘Ÿ NF (ยฎ๐‘ง 0 ) = 2 ๐‘Ÿ NF,๐‘– (ยฎ๐‘ง0 ). (6.132) ๐‘– This definition of the (overall) normal form radius corresponds to the following definition for the (overall) normal form defect ๐‘‘NF (ยฎ๐‘ง0 ) = ๐‘Ÿ NF ( M (ยฎ๐‘ง0 )) โˆ’ ๐‘Ÿ NF (ยฎ๐‘ง0 ) . (6.133) Unless stated otherwise, we will be using and referring to the (overall) normal form radius and the (overall) normal form defect. 6.3.4 The Search Domain in the Form of Onion Layers The onion layer approach describes a way to partition the phase space regions and determine the associated variables for the verified global optimization. For the partitioning, it is important to 206 consider the dynamics of the system. In Chapter 5, we saw that the main characteristics of the phase space motion in the storage ring are the oscillation amplitudes and the momentum offset ๐›ฟ ๐‘. Accordingly, we want to calculate the verified stability estimates on the rate of divergence based on partitions categorized by those criteria. While the partitioning according to the momentum offset ๐›ฟ ๐‘ is straightforward, defining the partitions of different phase space amplitudes is not, because the phase space curve of a particle with a certain amplitude forms a nonlinearly distorted elliptical shape in the original phase space. The onion layer approach (see Fig. 6.25) partitions the phase space along those nonlinearly distorted elliptical phase space curves using the normal form transformation. 1.5 1 4 1 0.8 2 0.5 ๐‘Ž 0 ๐‘ 0 ๐‘Ÿ NF,2 0.6 โˆ’2 โˆ’0.5 0.4 โˆ’1 โˆ’4 0.2 โˆ’1.5 0 โˆ’40 โˆ’20 0 20 40 โˆ’40 โˆ’20 0 20 40 0 0.2 0.4 0.6 0.8 1 ๐‘ฅ ๐‘ฆ ๐‘Ÿ NF,1 Figure 6.25: The left and the middle plot show the representation of an onion layer (black region) in regular phase space coordinates. The thickness of the onion layer is determined by the range in ๐‘Ÿ NF,1 and ๐‘Ÿ NF,2 as well as the range in ๐›ฟ ๐‘. For this particular example, we set ๐›ฟ ๐‘ to a fixed value of ๐›ฟ ๐‘ = 0% instead of a range. The range in the normal form radii is given by ๐‘Ÿ NF,1 โˆˆ [0.15, 0.25] and ๐‘Ÿ NF,2 โˆˆ [0.7, 0.75]. Note that the thickness in ๐‘Ÿ NF,1 is twice the thickness in ๐‘Ÿ NF,2 . Accordingly, the projection of the onion layer into the radial phase space (๐‘ฅ, ๐‘Ž) appears roughly twice as thick as the projection into the vertical phase space (๐‘ฆ, ๐‘). As illustrated in Fig. 6.25, the normal form coordinates allow us to partition by amplitude. They are our best approximation of mapping the orbital phase space behavior onto circles. Accordingly, we can use the normal form description of the motion to define the onion layers for the global optimization. Specifically, we chose the normal form radii ๐‘Ÿ NF,1 and ๐‘Ÿ NF,2 as well as the corresponding normal form phase space angles ๐œ™NF,1 and ๐œ™NF,2 as the optimization variables. Additionally, the momentum offset ๐›ฟ ๐‘ is also considered an optimization variable. The normal form phase space variables (๐‘ž NF,1 , ๐‘ NF,1 ) and (๐‘ž NF,2 , ๐‘ NF,2 ) are expressed in 207 terms of the polar optimization variables with   ยฉ ๐‘ž NF,1 ยช ยฉcos ๐œ™NF,1 ยช ยฉ ๐‘ž NF,2 ยช ยฉcos ๐œ™NF,2 ยช ยญ ยญ ยฎ = ๐‘Ÿ NF,1 ยญ ยฎ ยญ ยฎ ยฎ and ยญ ยญ ยฎ = ๐‘Ÿ NF,2 ยญ ยฎ ยญ ยฎ. ยฎ (6.134) ๐‘ sin ๐œ™NF,1 ๐‘ sin ๐œ™NF,2 ยซ NF,1 ยฌ ยซ ยฌ ยซ NF,2 ยฌ ยซ ยฌ The inverse normal form transformation Aโˆ’1 is then used as a vehicle to express the relevant phase space regions in original phase space (๐‘ฅ, ๐‘Ž, ๐‘ฆ, ๐‘) in terms of the optimization variables (๐‘Ÿ NF,1 , ๐œ™NF,1 , ๐‘Ÿ NF,2 , ๐œ™NF,2 , ๐›ฟ ๐‘). Moving along the angles ๐œ™NF,1 and ๐œ™NF,2 will approximately move along the phase space curve in the original coordinates. Accordingly, the search domain in those optimization variables is always [โˆ’๐œ‹, ๐œ‹]. The domain on the normal form radii and the momentum offset determines the thickness of the onion layer, as illustrated in Fig. 6.25, and is set to 0.04 for normal form radii and to 0.04% in the momentum offset space. 6.3.5 The Complexity and Nonlinearity of the Normal Form Defect Function In Chapter 5, we analyzed the normal form defect that individual particles encounter during stroboscopic tracking. In other words, we only probed individual phase space points of a particleโ€™s orbit for its normal form defect. We found that muons that encounter phase space regions with larger normal form defects are more likely to get lost (see Fig. 5.12). However, the probing only yields an incomplete picture of the normal form defect that a particle can potentially encounter. Fig. 6.26 illustrates how much the normal form defect can vary for fixed normal form amplitudes that approximately represent the normal form defect landscape along the phase space curve of a single particle. Fig. 6.26 illustrates the normal form defect ๐‘‘NF,1 of an onion layer of zero thickness, which is given by a single point in the 3D onion layer thickness space of ๐‘Ÿ NF,1 , ๐‘Ÿ NF,2 , and ๐›ฟ ๐‘. The landscape is characterized by highly nonlinear behavior with many local minima and maxima, which are extreme points of very steep valleys and hills. Accordingly, the stroboscopic normal form defect probing while tracking can significantly underestimate the maximum normal form defect of an orbit 208 8.0E-8 6.0E-8 4.0E-8 2.0E-8 0 -2.0E-8 -4.0E-8 -6.0E-8 -8.0E-8 ๐œ‹ 0๐œ™NF,2 โˆ’๐œ‹ 0 ๐œ™NF,1 ๐œ‹ โˆ’๐œ‹ Figure 6.26: Normal form defect landscape of the radial phase space in ๐œ™NF,1 and ๐œ™NF,2 for fixed normal form amplitudes of ๐‘Ÿ NF,1 = 0.4 and ๐‘Ÿ NF,2 = 0.4, and with ๐›ฟ ๐‘ = 0%. The underlying map considers an ESQ voltage of 18.3 kV. in a certain phase space region, which motivates a rigorous analysis of the normal form defect for those phase space regions. In Fig. 6.27 and Fig. 6.28, the normal form defect landscapes in the vertical and radial direction are shown for maps considering an ESQ voltage of 18.3 kV and 17.5 kV, respectively. The different normal form defect landscapes emphasize how much the landscapes change in shape and magnitude for different normal form phase space points. 209 ๐œ‹ ๐œ‹ 2.0E-6 2.0E-7 1.5E-6 1.0E-6 1.0E-7 5.0E-7 ๐œ™NF,2 0 ๐œ™NF,2 0 0 0 -5.0E-7 -1.0E-7 -1.0E-6 -2.0E-7 -1.5E-6 โˆ’๐œ‹ โˆ’๐œ‹ โˆ’๐œ‹ 0 ๐œ‹ โˆ’๐œ‹ 0 ๐œ‹ ๐œ™NF,1 ๐œ™NF,1 ๐œ‹ ๐œ‹ 6.0E-7 3.0E-6 5.0E-7 4.0E-7 2.0E-6 3.0E-7 1.0E-6 ๐œ™NF,2 0 2.0E-7 ๐œ™NF,2 0 1.0E-7 0 0 -1.0E-6 -1.0E-7 -2.0E-7 -2.0E-6 โˆ’๐œ‹ โˆ’๐œ‹ โˆ’๐œ‹ 0 ๐œ‹ โˆ’๐œ‹ 0 ๐œ‹ ๐œ™NF,1 ๐œ™NF,1 ๐œ‹ 2.0E-4 ๐œ‹ 1.5E-4 1.0E-3 1.0E-4 5.0E-4 5.0E-5 ๐œ™NF,2 0 ๐œ™NF,2 0 0 0 -5.0E-5 -1.0E-4 -5.0E-4 -1.5E-4 โˆ’๐œ‹ โˆ’๐œ‹ -1.0E-3 โˆ’๐œ‹ 0 ๐œ‹ โˆ’๐œ‹ 0 ๐œ‹ ๐œ™NF,1 ๐œ™NF,1 Figure 6.27: The normal form defect landscape of the radial (left side) and vertical (right side) phase space for multiple onion layers of zero thickness, which are characterized by (๐‘Ÿ NF,1 , ๐‘Ÿ NF,2 , ๐›ฟ ๐‘). The top row corresponds to (0.1, 0.2, 0.24%), the middle row corresponds to (0.2, 0.05, 0.24%), and the bottom row corresponds to (0.56, 0.72, 0.04%). The underlying map considers an ESQ voltage of 18.3 kV. 210 ๐œ‹ ๐œ‹ 2.0E-6 2.0E-7 1.5E-6 1.0E-6 1.0E-7 5.0E-7 ๐œ™NF,2 0 ๐œ™NF,2 0 0 0 -5.0E-7 -1.0E-7 -1.0E-6 -2.0E-7 -1.5E-6 โˆ’๐œ‹ โˆ’๐œ‹ โˆ’๐œ‹ 0 ๐œ‹ โˆ’๐œ‹ 0 ๐œ‹ ๐œ™NF,1 ๐œ™NF,1 ๐œ‹ ๐œ‹ 6.0E-7 3.0E-6 5.0E-7 4.0E-7 2.0E-6 3.0E-7 1.0E-6 ๐œ™NF,2 0 2.0E-7 ๐œ™NF,2 0 1.0E-7 0 0 -1.0E-6 -1.0E-7 -2.0E-7 -2.0E-6 โˆ’๐œ‹ โˆ’๐œ‹ โˆ’๐œ‹ 0 ๐œ‹ โˆ’๐œ‹ 0 ๐œ‹ ๐œ™NF,1 ๐œ™NF,1 ๐œ‹ 2.0E-4 ๐œ‹ 1.5E-4 1.0E-3 1.0E-4 5.0E-4 5.0E-5 ๐œ™NF,2 0 ๐œ™NF,2 0 0 0 -5.0E-5 -1.0E-4 -5.0E-4 -1.5E-4 โˆ’๐œ‹ โˆ’๐œ‹ -1.0E-3 โˆ’๐œ‹ 0 ๐œ‹ โˆ’๐œ‹ 0 ๐œ‹ ๐œ™NF,1 ๐œ™NF,1 Figure 6.28: The normal form defect landscape of the radial (left side) and vertical (right side) phase space for multiple onion layers of zero thickness, which are characterized by (๐‘Ÿ NF,1 , ๐‘Ÿ NF,2 , ๐›ฟ ๐‘). The top row corresponds to (0.1, 0.2, 0.24%), the middle row corresponds to (0.2, 0.05, 0.24%), and the bottom row corresponds to (0.56, 0.72, 0.04%). The underlying map considers an ESQ voltage of 17.5 kV. 211 Comparing the normal form defect of the radial and vertical phase space clearly shows the different orders of magnitude at play for those particular onion layers of zero thickness. The normal form defect of the vertical phase space is about 1.5 orders of magnitude larger than the normal form defect of the radial phase space. The comparison between Fig. 6.27 and Fig. 6.28 shows something rather fascinating. Even though the normal form defect landscapes change so drastically for different phase space positions, they are very similar for the two maps at the same normal form positions. The magnitude of the normal form defect is usually higher for the 18.3 kV, but the example in the bottom row shows that there are also normal form phase space regions where it is the other way around. The top row and middle row of Fig. 6.27 and Fig. 6.28 show phase space points with the same momentum offset and roughly the same overall normal form radius. While the magnitude of the normal form defects in the radial and vertical direction is roughly the same, the shape of the normal form defect landscape differs tremendously. For the global optimization, this means that the objective function looks vastly different for each of the onion layer search domains. 6.3.6 The Results of the Verified Global Optimization of the Normal Form Defect As mentioned in Sec. 6.3.4, we partition the search space into onion layers of the size 0.04 ร— 2๐œ‹ ร— 0.04 ร— 2๐œ‹ ร— 0.04% in (๐‘Ÿ NF,1 , ๐œ™NF,1 , ๐‘Ÿ NF,2 , ๐œ™NF,2 , ๐›ฟ ๐‘). Based on the ๐›ฟ ๐‘ rage of the realistic particle distribution in Chapter 5, we investigate the ๐›ฟ ๐‘ space from โˆ’0.22% to +0.42% in 16 partitions of size 0.04%. For each of those 16 pieces, we additionally partition the (๐‘Ÿ NF,1 , ๐‘Ÿ NF,2 ) space into boxes of size 0.04 ร— 0.04. To determine which of those boxes represent phase space behavior within the collimator region, we probe the bottom left corner of each box, namely, the point with the lowest normal form amplitudes (๐‘Ÿ NF,1,min , ๐‘Ÿ NF,2,min ) and check if those lowest amplitudes are already outside the collimator region in the original phase space coordinates. For the probing, we take 30 ร— 30 ร— 2 testing points in ๐œ™NF,1 , ๐œ™NF,2 , ๐›ฟ ๐‘ and map them back into the original phase space (๐‘ฅ, ๐‘Ž, ๐‘ฆ, ๐‘) using the inverse normal form transformation Aโˆ’1 . The two values for ๐›ฟ ๐‘ are the maximum and minimum 212 momentum offset of the onion layer. A box is only analyzed if all of the 1800 probing points satisfy โˆš๏ธ ๐‘ฅ 2 + ๐‘ฆ 2 < 0.045 mm. To benchmark the verified analysis, we also present a nonverified normal form defect analysis of the same onion layers. The nonverified analysis is based on probes of the top right corner of each box, namely, the point with the largest normal form amplitudes (๐‘Ÿ NF,1,max , ๐‘Ÿ NF,2,max ). The 30 ร— 30 ร— 2 probing points in ๐œ™NF,1 , ๐œ™NF,2 , ๐›ฟ ๐‘ are chosen the same way as above. This probing approach is used in the verified analysis as a method to obtain a good initial cutoff value for the verified global optimizer. Accordingly, the nonverified analysis provides a lower bound on the maximum normal form defect, while the verified analysis constitutes an upper bound. The results on the following pages (see Fig. 6.29 to Fig. 6.32) are ordered such that the nonverified probing analysis can be compared to the verified global optimization by switching back and forth between pages. Fig. 6.29 and Fig. 6.30 respectively show the nonverified and verified analysis for the map with an ESQ voltage of 17.5 kV, while Fig. 6.31 and Fig. 6.32 respectively show the verified and the nonverified analysis for the map with an ESQ voltage of 18.3 kV. Additionally, the two verified normal form defect analyses in Fig. 6.30 and Fig. 6.31 for the map with an ESQ voltage of 17.5 kV and 18.3 kV, respectively, can be compared the same way. The color scheme in Fig. 6.29 to Fig. 6.32 indicates the maximum normal form defect in each of the onion layers. Given the 0.04 ร— 0.04 box size of the onion layers in normal form space and the maximum normal form defect in the onion layer ๐‘‘NF,max , we can calculate the minimum number of turns ๐‘ required to cross through each onion layer as a Nekhoroshev-type stability estimate with 0.04 ๐‘= . (6.135) ๐‘‘NF,max The inner white onion layers have a maximum normal form defect below 10โˆ’5 . Accordingly, even in the worst case, it takes at least 4000 turns to cross the respective onion layer. It takes at least 400 turns to cross a yellow onion layer by the same measure and at least 40 turns to cross an orange onion layer. Red onion layers take at least 12 turns to cross, and black onion layers can, in the worst case, be crossed in fewer turns. 213 1 [โˆ’0.22%, โˆ’0.18%] [โˆ’0.18%, โˆ’0.14%] [โˆ’0.14%, โˆ’0.10%] [โˆ’0.10%, โˆ’0.06%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [โˆ’0.06%, โˆ’0.02%] [โˆ’0.02%, +0.02%] [+0.02%, +0.06%] [+0.06%, +0.10%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [+0.10%, +0.14%] [+0.14%, +0.18%] [+0.18%, +0.22%] [+0.22%, +0.26%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [+0.26%, +0.30%] [+0.30%, +0.34%] [+0.34%, +0.38%] [+0.38%, +0.42%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 Figure 6.29: Nonverified normal form defect for the phase space storage regions of the Muon ๐‘”-2 Storage Ring simulation with an ESQ voltage of 17.5 kV. The individual plots show different momentum ranges, clarified by the label at the top of each graph. The color scheme corresponds to the normal form defect of the specific onion layer. The white boxes for lower normal form radii indicate a normal form defect below 10โˆ’5 . The yellow boxes denote normal form defects up to 10โˆ’4 . The orange boxes correspond to normal form defects up to 10โˆ’3 . The red boxes denote normal form defects up to 10โˆ’2.5 , and the black boxes indicate normal form defects larger than that. Each onion layer corresponds to a 0.04 ร— 0.04 box in normal form space with a thickness of 0.04% in ๐›ฟ ๐‘. 214 1 [โˆ’0.22%, โˆ’0.18%] [โˆ’0.18%, โˆ’0.14%] [โˆ’0.14%, โˆ’0.10%] [โˆ’0.10%, โˆ’0.06%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [โˆ’0.06%, โˆ’0.02%] [โˆ’0.02%, +0.02%] [+0.02%, +0.06%] [+0.06%, +0.10%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [+0.10%, +0.14%] [+0.14%, +0.18%] [+0.18%, +0.22%] [+0.22%, +0.26%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [+0.26%, +0.30%] [+0.30%, +0.34%] [+0.34%, +0.38%] [+0.38%, +0.42%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 Figure 6.30: Verified normal form defect for the phase space storage regions of the Muon ๐‘”-2 Storage Ring simulation with an ESQ voltage of 17.5 kV. The individual plots show different momentum ranges, clarified by the label at the top of each graph. The color scheme corresponds to the normal form defect of the specific onion layer. The white boxes for lower normal form radii indicate a normal form defect below 10โˆ’5 . The yellow boxes denote normal form defects up to 10โˆ’4 . The orange boxes correspond to normal form defects up to 10โˆ’3 . The red boxes denote normal form defects up to 10โˆ’2.5 , and the black boxes indicate normal form defects larger than that. Each onion layer corresponds to a 0.04 ร— 0.04 box in normal form space with a thickness of 0.04% in ๐›ฟ ๐‘. 215 1 [โˆ’0.22%, โˆ’0.18%] [โˆ’0.18%, โˆ’0.14%] [โˆ’0.14%, โˆ’0.10%] [โˆ’0.10%, โˆ’0.06%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [โˆ’0.06%, โˆ’0.02%] [โˆ’0.02%, +0.02%] [+0.02%, +0.06%] [+0.06%, +0.10%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [+0.10%, +0.14%] [+0.14%, +0.18%] [+0.18%, +0.22%] [+0.22%, +0.26%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [+0.26%, +0.30%] [+0.30%, +0.34%] [+0.34%, +0.38%] [+0.38%, +0.42%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 Figure 6.31: Verified normal form defect for the phase space storage regions of the Muon ๐‘”-2 Storage ring simulation with an ESQ voltage of 18.3 kV. The individual plots show different momentum ranges, clarified by the label at the top of each graph. The color scheme corresponds to the normal form defect of the specific onion layer. The white boxes for lower normal form radii indicate a normal form defect below 10โˆ’5 . The yellow boxes denote normal form defects up to 10โˆ’4 . The orange boxes correspond to normal form defects up to 10โˆ’3 . The red boxes denote normal form defects up to 10โˆ’2.5 , and the black boxes indicate normal form defects larger than that. Each onion layer corresponds to a 0.04 ร— 0.04 box in normal form space with a thickness of 0.04% in ๐›ฟ ๐‘. 216 1 [โˆ’0.22%, โˆ’0.18%] [โˆ’0.18%, โˆ’0.14%] [โˆ’0.14%, โˆ’0.10%] [โˆ’0.10%, โˆ’0.06%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [โˆ’0.06%, โˆ’0.02%] [โˆ’0.02%, +0.02%] [+0.02%, +0.06%] [+0.06%, +0.10%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [+0.10%, +0.14%] [+0.14%, +0.18%] [+0.18%, +0.22%] [+0.22%, +0.26%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [+0.26%, +0.30%] [+0.30%, +0.34%] [+0.34%, +0.38%] [+0.38%, +0.42%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 Figure 6.32: Nonverified normal form defect for the phase space storage regions of the Muon ๐‘”-2 Storage Ring simulation with an ESQ voltage of 18.3 kV. The individual plots show different momentum ranges which are clarified by the label at the top of each graph. The color scheme corresponds to the normal form defect of the specific onion layer. The white boxes for lower normal form radii indicate a normal form defect below 10โˆ’5 . The yellow boxes denote normal form defects up to 10โˆ’4 . The orange boxes correspond to normal form defects up to 10โˆ’3 . The red boxes denote normal form defects up to 10โˆ’2.5 , and the black boxes indicate normal form defects larger than that. Each onion layer corresponds to a 0.04 ร— 0.04 box in normal form space with a thickness of 0.04% in ๐›ฟ ๐‘. 217 6.3.7 Comparison of Verified Nekhoroshev-type Stability Estimates to Actual Rates of Divergence in Example Island Structure The minimum turn numbers of the verified Nekhoroshev-type stability estimates are a verified underestimation of the minimum number of turns it takes particles to cross a respective onion layer. The estimation assumes that the maximum normal form defect of the onion layer is encountered in every turn. To put this underestimation in perspective we take a look at the island patterns from Fig. 5.30 in the storage ring configuration with an ESQ voltage of 18.3 kV. The particles tracked in Fig. 5.30 all have a momentum offset of ๐›ฟ ๐‘ = 0.126% and only differ in their vertical amplitude โ€“ the radial amplitude is constant with ๐‘Ÿ NF,1 โ‰ˆ 0. We number the five islands from smallest (1) to largest (5). In Tab. 6.16, the minimal and maximal normal form radii of each island are listed together with the number of turns it takes the islands to get from their lowest normal form amplitude to the largest. The number of turns is directly related to the period at which the vertical amplitude of the particles is modulated due to the island structure. The normal form radius range divided by the number of turns yields the average normal form defect of the particle. Table 6.16: Analysis of the normal form radius range of the five island particles in Fig. 5.30 and the number of turns it requires to get from the lower end of the range to the upper end. The islands are numbered from smallest (1) to largest (5). island ๐‘Ÿ NF,2,min ๐‘Ÿ NF,2,max turns avg. ๐‘‘NF (1) 0.925 0.932 244 2.8E-5 (2) 0.872 0.983 254 4.4E-4 (3) 0.820 1.028 293 7.1E-4 (4) 0.793 1.051 347 7.4E-4 (5) 0.773 1.066 666 4.4E-4 For the small islands close to the period-3 fixed points, the number of turns required to get from the lowest to the highest normal form amplitude increases only slightly with the size of the island. As a consequence, the average normal form defect increases. For the very large islands, the relation is quite the opposite. The range gets only slightly larger, but the period of modulation increases rapidly such that the average normal form defect even decreases again. 218 Tab. 6.17 lists the number of turns it takes the various island particles to cross the onion layers [0.84, 0.88], [0.88, 0.92], [0.92, 0.96], and [0.96, 1] in ๐‘Ÿ NF,2 together with the predicted minimum number of turns required to cross the onion layer provided by the verified analysis. Table 6.17: Number of turns required by the islands from Fig. 5.30 to cross the given onion layers in ๐‘Ÿ NF,2 direction. The islands are numbered from smallest (1) to largest (5). Additionally, the minimum number of turns required to cross the onion layer determined by the verified analysis is shown. [0.84, 0.88] [0.88, 0.92] [0.92, 0.96] [0.96, 1.00] island (1) - - - - island (2) - 81 58 - island (3) 65 40 32 30 island (4) 59 36 27 23 island (5) 104 36 25 21 Verified Analysis >15.7 >8.3 >4.5 >2.5 The dynamics in a single onion layer like [0, 0.04] ร— [0.92, 0.96] ร— [0.10%, 0.14%] can vary significantly. Some orbits remain in an onion layer indefinitely, like the smallest island (1). In contrast, others are transported through it with sometimes less than a factor ten between the worst case divergence predicted by the verified normal form defect analysis and the actual rate of divergence. The analysis of the largest island (5) is particularly interesting, because the average rate of divergence varies quite significantly over the normal form radius range. In short, it is possible to relate the quantitative aspects of the normal form defect analysis to the actual dynamics within the onion layer, and in particular, the potentially worst case dynamics. 6.3.8 Relevance of the ESQ Voltage on the Stability The global normal form defect analysis is also very powerful for the qualitative stability analysis of different storage ring configurations. A comparison of Fig. 6.30 and Fig. 6.31 yields obvious differences between the verified normal form defect of the map with an ESQ voltage of 17.5 kV and the map with an ESQ voltage of 18.3 kV. There are clearly more diverging regions with a larger maximum normal form defect for 18.3 kV in Fig. 6.31 than there are for the 17.5 kV map in Fig. 6.30. 219 Note that the individual onion layers of the 18.3 kV map in Fig. 6.31 and the 17.5 kV map in Fig. 6.30 do not necessarily correspond to the same phase space regions in the (๐‘ฅ, ๐‘Ž, ๐‘ฆ, ๐‘) phase space. The normal form transformation of each map is slightly different such that the representation of the relevant (๐‘ฅ, ๐‘Ž, ๐‘ฆ, ๐‘) phase space in normal form space can be different for the two maps. However, each of the 16 plots show the exact same viable (๐‘ฅ, ๐‘Ž, ๐‘ฆ, ๐‘) phase space in the normal form coordinates just with a slightly different scaling in ๐‘Ÿ NF,1 and ๐‘Ÿ NF,2 . Accordingly, comparing the color distributions for each of the 16 plots between the two maps is a valid measure to compare the stability of the two storage ring configurations. As previously mentioned, this more diverging behavior of the map with an ESQ voltage of 18.3 kV compared to the map with an ESQ voltage of 17.5 kV seen in Fig. 6.31 compared to Fig. 6.30, respectively, is very likely linked to the closeness of low-order resonances and their associated fixed point structures and the resulting amplitude modulations. Specifically, we saw in Chapter 5 that the vertical 1/3-resonance tune and its associated period-3 fixed point structures for the simulation using an ESQ voltage of 18.3 kV were a major loss and instability factor. To illustrate the difference in the closeness to low-order resonances, Fig. 6.33 to Fig. 6.35 show the tune shifts of the 17.5 kV map. The tune shifts are of similar magnitude and complexity as the tune shifts of the 18.3 kV map previously shown in Chapter 5 in Fig. 5.7 to Fig. 5.9. However, the absolute values of the tunes for 17.5 kV are in lower vertical tune ranges and therefore further away from the vertical low-order 1/3 resonance tune. Even under the combined influence of both the radial and vertical amplitude, as well as the momentum offset, none of the tunes of the 17.5 kV map cross the vertical 1/3 resonance tune. In contrast, almost for every momentum offset there is a combination of radial and vertical amplitudes that crosses the vertical 1/3 resonance tune for the 18.3 kV map. This suggests that there are no period-3 fixed point structures within the storage region, which would explain the less diverging onion layer picture in Fig. 6.30 compared to Fig. 6.31. In summary, both the tune analysis as well as the normal form defect analysis could show that the map with an ESQ voltage of 18.3 kV yields more potential diverging behavior and instability. 220 ๐‘ฅamp 0.96 ๐›ฟ ๐‘ = 0.40% 0.335 0.955 0.33 0.325 45 0.95 ๐œˆ๐‘ฅ ๐œˆ๐‘ฆ 0.32 0.945 0.315 0.94 40 0.31 0.935 0.305 ๐›ฟ ๐‘ = 0.40% 0.96 ๐›ฟ ๐‘ = 0.32% 0.335 35 0.955 0.33 0.325 0.95 30 ๐œˆ๐‘ฅ ๐œˆ๐‘ฆ 0.32 0.945 0.315 0.94 0.31 25 0.935 0.305 ๐›ฟ ๐‘ = 0.32% 0.96 ๐›ฟ ๐‘ = 0.26% 0.335 20 0.955 0.33 0.325 0.95 ๐œˆ๐‘ฅ ๐œˆ๐‘ฆ 0.32 15 0.945 0.315 0.94 0.31 10 0.935 0.305 ๐›ฟ ๐‘ = 0.26% 0.96 ๐›ฟ ๐‘ = 0.14% 0.335 5 0.955 0.33 0.325 0.95 ๐œˆ๐‘ฅ ๐œˆ๐‘ฆ 0.32 0.945 0 0.315 0.94 0.31 0.935 0.305 ๐›ฟ ๐‘ = 0.14% 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 ๐‘ฆ amp [mm] ๐‘ฆ amp [mm] Figure 6.33: Behavior of combined amplitude dependent tune shifts at multiple momentum offsets for an ESQ voltage of 17.5 kV. 221 ๐‘ฅamp 0.96 ๐›ฟ ๐‘ = 0.40% 0.335 0.955 0.33 0.325 45 0.95 ๐œˆ๐‘ฅ ๐œˆ๐‘ฆ 0.32 0.945 0.315 0.94 40 0.31 0.935 0.305 ๐›ฟ ๐‘ = 0.40% 0.96 ๐›ฟ ๐‘ = 0.32% 0.335 35 0.955 0.33 0.325 0.95 30 ๐œˆ๐‘ฅ ๐œˆ๐‘ฆ 0.32 0.945 0.315 0.94 0.31 25 0.935 0.305 ๐›ฟ ๐‘ = 0.32% 0.96 ๐›ฟ ๐‘ = 0.26% 0.335 20 0.955 0.33 0.325 0.95 ๐œˆ๐‘ฅ ๐œˆ๐‘ฆ 0.32 15 0.945 0.315 0.94 0.31 10 0.935 0.305 ๐›ฟ ๐‘ = 0.26% 0.96 ๐›ฟ ๐‘ = 0.14% 0.335 5 0.955 0.33 0.325 0.95 ๐œˆ๐‘ฅ ๐œˆ๐‘ฆ 0.32 0.945 0 0.315 0.94 0.31 0.935 0.305 ๐›ฟ ๐‘ = 0.14% 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 ๐‘ฆ amp [mm] ๐‘ฆ amp [mm] Figure 6.34: Behavior of combined amplitude dependent tune shifts at multiple momentum offsets for an ESQ voltage of 17.5 kV. 222 ๐‘ฅamp 0.96 ๐›ฟ ๐‘ = 0.40% 0.335 0.955 0.33 0.325 45 0.95 ๐œˆ๐‘ฅ ๐œˆ๐‘ฆ 0.32 0.945 0.315 0.94 40 0.31 0.935 0.305 ๐›ฟ ๐‘ = 0.40% 0.96 ๐›ฟ ๐‘ = 0.32% 0.335 35 0.955 0.33 0.325 0.95 30 ๐œˆ๐‘ฅ ๐œˆ๐‘ฆ 0.32 0.945 0.315 0.94 0.31 25 0.935 0.305 ๐›ฟ ๐‘ = 0.32% 0.96 ๐›ฟ ๐‘ = 0.26% 0.335 20 0.955 0.33 0.325 0.95 ๐œˆ๐‘ฅ ๐œˆ๐‘ฆ 0.32 15 0.945 0.315 0.94 0.31 10 0.935 0.305 ๐›ฟ ๐‘ = 0.26% 0.96 ๐›ฟ ๐‘ = 0.14% 0.335 5 0.955 0.33 0.325 0.95 ๐œˆ๐‘ฅ ๐œˆ๐‘ฆ 0.32 0.945 0 0.315 0.94 0.31 0.935 0.305 ๐›ฟ ๐‘ = 0.14% 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 ๐‘ฆ amp [mm] ๐‘ฆ amp [mm] Figure 6.35: Behavior of combined amplitude dependent tune shifts at multiple momentum offsets for an ESQ voltage of 17.5 kV. 223 6.3.9 Comparison of Nonverified and Verified Normal Form Defect Analysis The differences between the nonverified and verified computations in Fig. 6.29 and Fig. 6.30 for an ESQ voltage of 17.5 kV, and Fig. 6.31 and Fig. 6.32 for an ESQ voltage of 18.3 kV are small but visible if one switches back and forth between the pages. To emphasize the differences between the verifed and nonverified computations onion layer by onion layer, Fig. 6.36 and Fig. 6.37 illustrate those differences for 17.5 kV and 18.3 kV, respectively. The differences show the importance of a verified method to capture each onion layerโ€™s maximum normal form defect, especially for the more diverging regions. To show that this difference is not an artifact of the bounding range of the global optimizer, Fig. 6.38 and Fig. 6.39 illustrate the difference between the optimized upper and the lower bound on the maximum normal form defect for the 17.5 kV map and 18.3 kV map, respectively. Because both figures only consist of white boxes, those differences are all smaller than 1E-5 and therefore do not alter the calculation of the differences between the nonverified and verified evaluation in Fig. 6.36 and Fig. 6.37 224 1 [โˆ’0.22%, โˆ’0.18%] [โˆ’0.18%, โˆ’0.14%] [โˆ’0.14%, โˆ’0.10%] [โˆ’0.10%, โˆ’0.06%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [โˆ’0.06%, โˆ’0.02%] [โˆ’0.02%, +0.02%] [+0.02%, +0.06%] [+0.06%, +0.10%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [+0.10%, +0.14%] [+0.14%, +0.18%] [+0.18%, +0.22%] [+0.22%, +0.26%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [+0.26%, +0.30%] [+0.30%, +0.34%] [+0.34%, +0.38%] [+0.38%, +0.42%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 Figure 6.36: Difference between verified normal form defect analysis and nonverified normal form defect analysis for the phase space storage regions of the Muon ๐‘”-2 Storage Ring simulation with an ESQ voltage of 17.5 kV. The individual plots show different momentum ranges which are clarified by the label at the top of each graph. The color scheme corresponds to the difference of the evaluated normal form defects of the specific onion layer. The white boxes for lower normal form radii indicate differences below 10โˆ’5 . The yellow boxes denote differences up to 10โˆ’4 , the orange boxes correspond to differences up to 10โˆ’3 , the red boxes denote differences up to 10โˆ’2.5 and the black boxes indicate differences larger than that. Each onion layer corresponds to a 0.04 ร— 0.04 box in normal form space with a thickness of 0.04% in ๐›ฟ ๐‘. 225 1 [โˆ’0.22%, โˆ’0.18%] [โˆ’0.18%, โˆ’0.14%] [โˆ’0.14%, โˆ’0.10%] [โˆ’0.10%, โˆ’0.06%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [โˆ’0.06%, โˆ’0.02%] [โˆ’0.02%, +0.02%] [+0.02%, +0.06%] [+0.06%, +0.10%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [+0.10%, +0.14%] [+0.14%, +0.18%] [+0.18%, +0.22%] [+0.22%, +0.26%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [+0.26%, +0.30%] [+0.30%, +0.34%] [+0.34%, +0.38%] [+0.38%, +0.42%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 Figure 6.37: Difference between verified normal form defect analysis and nonverified normal form defect analysis for the phase space storage regions of the Muon ๐‘”-2 Storage Ring simulation with an ESQ voltage of 18.3 kV. The individual plots show different momentum ranges, clarified by the label at the top of each graph. The color scheme corresponds to the difference of the evaluated normal form defects of the specific onion layer. The white boxes for lower normal form radii indicate a difference below 10โˆ’5 . The yellow boxes denote differences up to 10โˆ’4 . The orange boxes correspond to differences up to 10โˆ’3 . The red boxes denote differences up to 10โˆ’2.5 , and the black boxes indicate differences larger than that. Each onion layer corresponds to a 0.04 ร— 0.04 box in normal form space with a thickness of 0.04% in ๐›ฟ ๐‘. 226 1 [โˆ’0.22%, โˆ’0.18%] [โˆ’0.18%, โˆ’0.14%] [โˆ’0.14%, โˆ’0.10%] [โˆ’0.10%, โˆ’0.06%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [โˆ’0.06%, โˆ’0.02%] [โˆ’0.02%, +0.02%] [+0.02%, +0.06%] [+0.06%, +0.10%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [+0.10%, +0.14%] [+0.14%, +0.18%] [+0.18%, +0.22%] [+0.22%, +0.26%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [+0.26%, +0.30%] [+0.30%, +0.34%] [+0.34%, +0.38%] [+0.38%, +0.42%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 Figure 6.38: Difference between the rigorously guaranteed upper bound and the lower bound of the maximum normal form defect using Taylor Model based verified global optimization. The analysis is for the phase space storage regions of the Muon ๐‘”-2 Storage Ring simulation with an ESQ voltage of 17.5 kV. The individual plots show different momentum ranges which are clarified by the label at the top of each graph. The color scheme corresponds to the difference between the upper bound and the lower bound of the maximum normal form defect of the specific onion layer. All boxes are white because the difference is below 10โˆ’5 . Each onion layer corresponds to a 0.04 ร— 0.04 box in normal form space with a thickness of 0.04% in ๐›ฟ ๐‘. 227 1 [โˆ’0.22%, โˆ’0.18%] [โˆ’0.18%, โˆ’0.14%] [โˆ’0.14%, โˆ’0.10%] [โˆ’0.10%, โˆ’0.06%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [โˆ’0.06%, โˆ’0.02%] [โˆ’0.02%, +0.02%] [+0.02%, +0.06%] [+0.06%, +0.10%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [+0.10%, +0.14%] [+0.14%, +0.18%] [+0.18%, +0.22%] [+0.22%, +0.26%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [+0.26%, +0.30%] [+0.30%, +0.34%] [+0.34%, +0.38%] [+0.38%, +0.42%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 Figure 6.39: Difference between the rigorously guaranteed upper bound and the lower bound of the maximum normal form defect using Taylor Model based verified global optimization. The analysis is for the phase space storage regions of the Muon ๐‘”-2 Storage Ring simulation with an ESQ voltage of 18.3 kV. The individual plots show different momentum ranges which are clarified by the label at the top of each graph. The color scheme corresponds to the difference between the upper bound and the lower bound of the maximum normal form defect of the specific onion layer. All boxes are white because the difference is below 10โˆ’5 . Each onion layer corresponds to a 0.04 ร— 0.04 box in normal form space with a thickness of 0.04% in ๐›ฟ ๐‘. 228 6.3.10 The Analysis of the Effect of Normal Form Transformations of Different Order on the Normal Form Defect We use the normal form transformation as a function that provides pseudo-invariants of the motion, i.e., the normal form radii. By using the normal form transformation up to different orders, we can analyze the influence of the respective map orders on the dynamics of the system. In Fig. 6.40 to Fig. 6.49, the nonverified normal form defect analysis is performed for the tenth order map with a ESQ voltage of 18.3 kV using normal form transformations from order one to order ten. The normal form defect pictures for a normal form transformation of order five, six, and seven look identical even when carefully switching between pages. The largest improvement occurs with the ninth order normal form transformation because it captures large parts of the strong ninth order nonlinearities of the map caused by the 20th-pole of the ESQ potential. To further analyze if the tenth order map does indeed capture most of the relevant dynamics, we produce an eleventh order map and calculate its normal form defect using the tenth order normal form transformation (see Fig. 6.50). This kind of order increasing analysis is known from nonverified integrators with step size control. Compared to the tenth order map evaluation with the tenth order normal form transformation in Fig. 6.49, the eleventh order of the map leads to no visible difference, which is a good sign and suggests that a tenth order map is sufficient to capture the critical dynamics. However, this heuristic approach cannot guarantee that even higher order maps would also not yield a significant change. To capture this uncertainty of unknown higher order terms a verified map is required that includes all higher order errors in its Taylor Model remainder bound. 229 1 [โˆ’0.22%, โˆ’0.18%] [โˆ’0.18%, โˆ’0.14%] [โˆ’0.14%, โˆ’0.10%] [โˆ’0.10%, โˆ’0.06%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [โˆ’0.06%, โˆ’0.02%] [โˆ’0.02%, +0.02%] [+0.02%, +0.06%] [+0.06%, +0.10%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [+0.10%, +0.14%] [+0.14%, +0.18%] [+0.18%, +0.22%] [+0.22%, +0.26%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [+0.26%, +0.30%] [+0.30%, +0.34%] [+0.34%, +0.38%] [+0.38%, +0.42%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 Figure 6.40: Nonverified normal form defect for the phase space storage regions of the Muon ๐‘”-2 Storage Ring simulation with an ESQ voltage of 18.3 kV using the normal form transformation up to order 1 instead of the full tenth order. The individual plots show different momentum ranges, clarified by the label at the top of each graph. The color scheme corresponds to the normal form defect of the specific onion layer. The white boxes for lower normal form radii indicate a normal form defect below 10โˆ’5 . The yellow boxes denote normal form defects up to 10โˆ’4 . The orange boxes correspond to normal form defects up to 10โˆ’3 . The red boxes denote normal form defects up to 10โˆ’2.5 , and the black boxes indicate normal form defects larger than that. Each onion layer corresponds to a 0.04 ร— 0.04 box in normal form space with a thickness of 0.04% in ๐›ฟ ๐‘. 230 1 [โˆ’0.22%, โˆ’0.18%] [โˆ’0.18%, โˆ’0.14%] [โˆ’0.14%, โˆ’0.10%] [โˆ’0.10%, โˆ’0.06%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [โˆ’0.06%, โˆ’0.02%] [โˆ’0.02%, +0.02%] [+0.02%, +0.06%] [+0.06%, +0.10%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [+0.10%, +0.14%] [+0.14%, +0.18%] [+0.18%, +0.22%] [+0.22%, +0.26%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [+0.26%, +0.30%] [+0.30%, +0.34%] [+0.34%, +0.38%] [+0.38%, +0.42%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 Figure 6.41: Nonverified normal form defect for the phase space storage regions of the Muon ๐‘”-2 Storage Ring simulation with an ESQ voltage of 18.3 kV using the normal form transformation up to order 2 instead of the full tenth order. The individual plots show different momentum ranges, clarified by the label at the top of each graph. The color scheme corresponds to the normal form defect of the specific onion layer. The white boxes for lower normal form radii indicate a normal form defect below 10โˆ’5 . The yellow boxes denote normal form defects up to 10โˆ’4 . The orange boxes correspond to normal form defects up to 10โˆ’3 . The red boxes denote normal form defects up to 10โˆ’2.5 , and the black boxes indicate normal form defects larger than that. Each onion layer corresponds to a 0.04 ร— 0.04 box in normal form space with a thickness of 0.04% in ๐›ฟ ๐‘. 231 1 [โˆ’0.22%, โˆ’0.18%] [โˆ’0.18%, โˆ’0.14%] [โˆ’0.14%, โˆ’0.10%] [โˆ’0.10%, โˆ’0.06%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [โˆ’0.06%, โˆ’0.02%] [โˆ’0.02%, +0.02%] [+0.02%, +0.06%] [+0.06%, +0.10%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [+0.10%, +0.14%] [+0.14%, +0.18%] [+0.18%, +0.22%] [+0.22%, +0.26%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [+0.26%, +0.30%] [+0.30%, +0.34%] [+0.34%, +0.38%] [+0.38%, +0.42%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 Figure 6.42: Nonverified normal form defect for the phase space storage regions of the Muon ๐‘”-2 Storage Ring simulation with an ESQ voltage of 18.3 kV using the normal form transformation up to order 3 instead of the full tenth order. The individual plots show different momentum ranges, clarified by the label at the top of each graph. The color scheme corresponds to the normal form defect of the specific onion layer. The white boxes for lower normal form radii indicate a normal form defect below 10โˆ’5 . The yellow boxes denote normal form defects up to 10โˆ’4 . The orange boxes correspond to normal form defects up to 10โˆ’3 . The red boxes denote normal form defects up to 10โˆ’2.5 , and the black boxes indicate normal form defects larger than that. Each onion layer corresponds to a 0.04 ร— 0.04 box in normal form space with a thickness of 0.04% in ๐›ฟ ๐‘. 232 1 [โˆ’0.22%, โˆ’0.18%] [โˆ’0.18%, โˆ’0.14%] [โˆ’0.14%, โˆ’0.10%] [โˆ’0.10%, โˆ’0.06%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [โˆ’0.06%, โˆ’0.02%] [โˆ’0.02%, +0.02%] [+0.02%, +0.06%] [+0.06%, +0.10%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [+0.10%, +0.14%] [+0.14%, +0.18%] [+0.18%, +0.22%] [+0.22%, +0.26%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [+0.26%, +0.30%] [+0.30%, +0.34%] [+0.34%, +0.38%] [+0.38%, +0.42%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 Figure 6.43: Nonverified normal form defect for the phase space storage regions of the Muon ๐‘”-2 Storage Ring simulation with an ESQ voltage of 18.3 kV using the normal form transformation up to order 4 instead of the full tenth order. The individual plots show different momentum ranges, clarified by the label at the top of each graph. The color scheme corresponds to the normal form defect of the specific onion layer. The white boxes for lower normal form radii indicate a normal form defect below 10โˆ’5 . The yellow boxes denote normal form defects up to 10โˆ’4 . The orange boxes correspond to normal form defects up to 10โˆ’3 . The red boxes denote normal form defects up to 10โˆ’2.5 , and the black boxes indicate normal form defects larger than that. Each onion layer corresponds to a 0.04 ร— 0.04 box in normal form space with a thickness of 0.04% in ๐›ฟ ๐‘. 233 1 [โˆ’0.22%, โˆ’0.18%] [โˆ’0.18%, โˆ’0.14%] [โˆ’0.14%, โˆ’0.10%] [โˆ’0.10%, โˆ’0.06%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [โˆ’0.06%, โˆ’0.02%] [โˆ’0.02%, +0.02%] [+0.02%, +0.06%] [+0.06%, +0.10%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [+0.10%, +0.14%] [+0.14%, +0.18%] [+0.18%, +0.22%] [+0.22%, +0.26%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [+0.26%, +0.30%] [+0.30%, +0.34%] [+0.34%, +0.38%] [+0.38%, +0.42%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 Figure 6.44: Nonverified normal form defect for the phase space storage regions of the Muon ๐‘”-2 Storage Ring simulation with an ESQ voltage of 18.3 kV using the normal form transformation up to order 5 instead of the full tenth order. The individual plots show different momentum ranges, clarified by the label at the top of each graph. The color scheme corresponds to the normal form defect of the specific onion layer. The white boxes for lower normal form radii indicate a normal form defect below 10โˆ’5 . The yellow boxes denote normal form defects up to 10โˆ’4 . The orange boxes correspond to normal form defects up to 10โˆ’3 . The red boxes denote normal form defects up to 10โˆ’2.5 , and the black boxes indicate normal form defects larger than that. Each onion layer corresponds to a 0.04 ร— 0.04 box in normal form space with a thickness of 0.04% in ๐›ฟ ๐‘. 234 1 [โˆ’0.22%, โˆ’0.18%] [โˆ’0.18%, โˆ’0.14%] [โˆ’0.14%, โˆ’0.10%] [โˆ’0.10%, โˆ’0.06%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [โˆ’0.06%, โˆ’0.02%] [โˆ’0.02%, +0.02%] [+0.02%, +0.06%] [+0.06%, +0.10%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [+0.10%, +0.14%] [+0.14%, +0.18%] [+0.18%, +0.22%] [+0.22%, +0.26%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [+0.26%, +0.30%] [+0.30%, +0.34%] [+0.34%, +0.38%] [+0.38%, +0.42%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 Figure 6.45: Nonverified normal form defect for the phase space storage regions of the Muon ๐‘”-2 Storage Ring simulation with an ESQ voltage of 18.3 kV using the normal form transformation up to order 6 instead of the full tenth order. The individual plots show different momentum ranges, clarified by the label at the top of each graph. The color scheme corresponds to the normal form defect of the specific onion layer. The white boxes for lower normal form radii indicate a normal form defect below 10โˆ’5 . The yellow boxes denote normal form defects up to 10โˆ’4 . The orange boxes correspond to normal form defects up to 10โˆ’3 . The red boxes denote normal form defects up to 10โˆ’2.5 , and the black boxes indicate normal form defects larger than that. Each onion layer corresponds to a 0.04 ร— 0.04 box in normal form space with a thickness of 0.04% in ๐›ฟ ๐‘. 235 1 [โˆ’0.22%, โˆ’0.18%] [โˆ’0.18%, โˆ’0.14%] [โˆ’0.14%, โˆ’0.10%] [โˆ’0.10%, โˆ’0.06%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [โˆ’0.06%, โˆ’0.02%] [โˆ’0.02%, +0.02%] [+0.02%, +0.06%] [+0.06%, +0.10%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [+0.10%, +0.14%] [+0.14%, +0.18%] [+0.18%, +0.22%] [+0.22%, +0.26%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [+0.26%, +0.30%] [+0.30%, +0.34%] [+0.34%, +0.38%] [+0.38%, +0.42%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 Figure 6.46: Nonverified normal form defect for the phase space storage regions of the Muon ๐‘”-2 Storage Ring simulation with an ESQ voltage of 18.3 kV using the normal form transformation up to order 7 instead of the full tenth order. The individual plots show different momentum ranges, clarified by the label at the top of each graph. The color scheme corresponds to the normal form defect of the specific onion layer. The white boxes for lower normal form radii indicate a normal form defect below 10โˆ’5 . The yellow boxes denote normal form defects up to 10โˆ’4 . The orange boxes correspond to normal form defects up to 10โˆ’3 . The red boxes denote normal form defects up to 10โˆ’2.5 , and the black boxes indicate normal form defects larger than that. Each onion layer corresponds to a 0.04 ร— 0.04 box in normal form space with a thickness of 0.04% in ๐›ฟ ๐‘. 236 1 [โˆ’0.22%, โˆ’0.18%] [โˆ’0.18%, โˆ’0.14%] [โˆ’0.14%, โˆ’0.10%] [โˆ’0.10%, โˆ’0.06%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [โˆ’0.06%, โˆ’0.02%] [โˆ’0.02%, +0.02%] [+0.02%, +0.06%] [+0.06%, +0.10%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [+0.10%, +0.14%] [+0.14%, +0.18%] [+0.18%, +0.22%] [+0.22%, +0.26%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [+0.26%, +0.30%] [+0.30%, +0.34%] [+0.34%, +0.38%] [+0.38%, +0.42%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 Figure 6.47: Nonverified normal form defect for the phase space storage regions of the Muon ๐‘”-2 Storage Ring simulation with an ESQ voltage of 18.3 kV using the normal form transformation up to order 8 instead of the full tenth order. The individual plots show different momentum ranges, clarified by the label at the top of each graph. The color scheme corresponds to the normal form defect of the specific onion layer. The white boxes for lower normal form radii indicate a normal form defect below 10โˆ’5 . The yellow boxes denote normal form defects up to 10โˆ’4 . The orange boxes correspond to normal form defects up to 10โˆ’3 . The red boxes denote normal form defects up to 10โˆ’2.5 , and the black boxes indicate normal form defects larger than that. Each onion layer corresponds to a 0.04 ร— 0.04 box in normal form space with a thickness of 0.04% in ๐›ฟ ๐‘. 237 1 [โˆ’0.22%, โˆ’0.18%] [โˆ’0.18%, โˆ’0.14%] [โˆ’0.14%, โˆ’0.10%] [โˆ’0.10%, โˆ’0.06%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [โˆ’0.06%, โˆ’0.02%] [โˆ’0.02%, +0.02%] [+0.02%, +0.06%] [+0.06%, +0.10%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [+0.10%, +0.14%] [+0.14%, +0.18%] [+0.18%, +0.22%] [+0.22%, +0.26%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [+0.26%, +0.30%] [+0.30%, +0.34%] [+0.34%, +0.38%] [+0.38%, +0.42%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 Figure 6.48: Nonverified normal form defect for the phase space storage regions of the Muon ๐‘”-2 Storage Ring simulation with an ESQ voltage of 18.3 kV using the normal form transformation up to order 9 instead of the full tenth order. The individual plots show different momentum ranges, clarified by the label at the top of each graph. The color scheme corresponds to the normal form defect of the specific onion layer. The white boxes for lower normal form radii indicate a normal form defect below 10โˆ’5 . The yellow boxes denote normal form defects up to 10โˆ’4 . The orange boxes correspond to normal form defects up to 10โˆ’3 . The red boxes denote normal form defects up to 10โˆ’2.5 , and the black boxes indicate normal form defects larger than that. Each onion layer corresponds to a 0.04 ร— 0.04 box in normal form space with a thickness of 0.04% in ๐›ฟ ๐‘. 238 1 [โˆ’0.22%, โˆ’0.18%] [โˆ’0.18%, โˆ’0.14%] [โˆ’0.14%, โˆ’0.10%] [โˆ’0.10%, โˆ’0.06%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [โˆ’0.06%, โˆ’0.02%] [โˆ’0.02%, +0.02%] [+0.02%, +0.06%] [+0.06%, +0.10%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [+0.10%, +0.14%] [+0.14%, +0.18%] [+0.18%, +0.22%] [+0.22%, +0.26%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [+0.26%, +0.30%] [+0.30%, +0.34%] [+0.34%, +0.38%] [+0.38%, +0.42%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 Figure 6.49: Nonverified normal form defect for the phase space storage regions of the Muon ๐‘”-2 Storage Ring simulation with an ESQ voltage of 18.3 kV using the normal form transformation up to the full tenth order. The individual plots show different momentum ranges, clarified by the label at the top of each graph. The color scheme corresponds to the normal form defect of the specific onion layer. The white boxes for lower normal form radii indicate a normal form defect below 10โˆ’5 . The yellow boxes denote normal form defects up to 10โˆ’4 . The orange boxes correspond to normal form defects up to 10โˆ’3 . The red boxes denote normal form defects up to 10โˆ’2.5 , and the black boxes indicate normal form defects larger than that. Each onion layer corresponds to a 0.04 ร— 0.04 box in normal form space with a thickness of 0.04% in ๐›ฟ ๐‘. 239 1 [โˆ’0.22%, โˆ’0.18%] [โˆ’0.18%, โˆ’0.14%] [โˆ’0.14%, โˆ’0.10%] [โˆ’0.10%, โˆ’0.06%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [โˆ’0.06%, โˆ’0.02%] [โˆ’0.02%, +0.02%] [+0.02%, +0.06%] [+0.06%, +0.10%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [+0.10%, +0.14%] [+0.14%, +0.18%] [+0.18%, +0.22%] [+0.22%, +0.26%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 1 [+0.26%, +0.30%] [+0.30%, +0.34%] [+0.34%, +0.38%] [+0.38%, +0.42%] 0.8 ๐‘Ÿ NF,2 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 ๐‘Ÿ NF,1 Figure 6.50: Nonverified normal form defect for the phase space storage regions of the Muon ๐‘”-2 Storage Ring simulation with an ESQ voltage of 18.3 kV using an eleventh order map and its normal form transformation up to tenth order. The individual plots show different momentum ranges, clarified by the label at the top of each graph. The color scheme corresponds to the normal form defect of the specific onion layer. The white boxes for lower normal form radii indicate a normal form defect below 10โˆ’5 . The yellow boxes denote normal form defects up to 10โˆ’4 . The orange boxes correspond to normal form defects up to 10โˆ’3 . The red boxes denote normal form defects up to 10โˆ’2.5 , and the black boxes indicate normal form defects larger than that. Each onion layer corresponds to a 0.04 ร— 0.04 box in normal form space with a thickness of 0.04% in ๐›ฟ ๐‘. 240 CHAPTER 7 CONCLUSION We investigated a diverse set of nonlinear systems using normal forms and rigorous differential algebra methods. The differential algebra framework implemented in COSY INFINITY served as the backbone of all the methods and techniques in this thesis. It allowed us to establish algorithms and solutions up to arbitrary order and with floating point accuracy. The basis of our analysis constituted map representations of the various systems based on the underlying equations of motion. These stroboscopic descriptions of the dynamics were expanded around a fixed point corresponding to an equilibrium state of the motion. Using Poincarรฉ projections, the dimensionality of the system was reduced to the essential components of the systemโ€™s dynamics. For the bounded motion problem in the zonal gravitational field of the Earth in Chapter 4, the motion was considered within a four dimensional Poincarรฉ surface capturing all ascending node states. In Chapter 5, the dynamics within the Muon ๐‘”-2 Storage Ring were analyzed in transverse cross sections of the storage ring at multiple azimuthal locations. The origin preserving maps were then analyzed using high order normal forms to calculate a description of the phase space dynamics that is rotationally invariant up to calculation order. In Chapter 3, the normal form algorithm was discussed in full detail using the illustrative example of the centrifugal governor. In this particular case, the normal form radii, which constitute the (pseudo-)invariants of the motion up to calculation order produced by the normal form algorithm, were directly related to the energy of the system up to calculation order. Additionally, the normal form produced high order functional descriptions of the period of oscillation of the centrifugal governor arms around their equilibrium angle depending on the amplitude of oscillation and changes in the rotation frequency of the governor. For the bounded motion problem, this rotational invariant representation of the phase space motion provided by the normal form was used to transform the system into action-angle like coordinates. This allowed us to average the bounded motion quantities while maintaining their 241 functional dependence on the constants of motion. DA inversion methods were then used to enforce the bounded motion conditions and produce parameterized descriptions of the constants of motion, which yielded entire continuous sets of bounded motion orbits. We illustrated that the resulting sets of orbits remained bounded for decades and far beyond the practically relevant distances of formation flying missions. Our approach can possibly be advanced to the fully gravitationally perturbed case. However, the associated break of the rotational symmetry makes this already complex system even more complex. The introduced longitudinal dependence and the loss of the angular momentum component as a constant of motion increase the dimensionality of the problem by two. Accordingly, pseudo-circular orbits of the full state are required to expand the fixed point map around. Only further research can answer if and how the approach can be adjusted to compensate for the loss of a known constant of motion and the increase in dimensionality. In our analysis of the dynamics in the Muon ๐‘”-2 Storage Ring in Chapter 5, we studied the oscillation frequencies of particles in the radial and vertical transverse direction also known as the betatron tunes. The normal form transformation allowed us to calculate the functional dependence of the tunes on the momentum offset of the particles and their amplitude of oscillation. A major insight of this investigation was that particles over the entire momentum offset range could cross the vertical 1/3-resonance frequency for certain vertical and radial amplitude combinations. This closeness to the low order resonance triggered intensive lost muon tracking studies, which revealed period-3 fixed point structures in the vertical phase space. Particles caught around those period-3 fixed points experienced significant vertical amplitude modulations, which drastically increased their risk of hitting a collimator and getting lost in the process. Throughout the analysis, the strong ninth order nonlinearities of the map caused by the 20th-pole of the ESQ potential were prominent. They could be found as eighth order dependencies in the amplitude and momentum dependent tune shifts and be visualized by the drastic change in the tune footprint when comparing eighth order to tenth order results. To further assess the stability of the Muon ๐‘”-2 Storage Ring rigorously, we utilized Taylor 242 Model based verified global optimization in Chapter 6. The abilities of Taylor Model based global optimization was presented using the objective functions of different example problems. The generalized Rosenbrock function served as an example to illustrate different effects that can sometimes influence the optimization including the dependency problem and the cluster effect. We illustrated that Taylor Models and their associated advanced bounding techniques could drastically suppress those effects compared to other commonly used approaches. The Lennard-Jones problem was used to illustrate the many intricacies that have to be solved for rigorous global optimization of some complex systems. While the Lennard-Jones problem is easily formulated, its formal description with optimization variables and bounding to a rigorous initial search domain are far from trivial. Our discussion of the problem also illustrated the struggle associated with not being able to exclude manifolds from the search domain for which the objective function is not defined. For the rigorous stability analysis of the Muon ๐‘”-2 Storage Ring, we calculated verified upper bounds on the rate at which particles can escape the storage region. To get a detailed understanding of the stability properties of the storage ring, we partitioned the five dimensional storage region into more than 8000 sections using the onion layer approach. We used Taylor Model based verified global optimization to calculate the maximum rate of divergence in the form of the normal form defect for each one of those partitions. The verified normal form defect results from the map with the closeness to the vertical 1/3-resonance from Chapter 5 were compared to the results of a map with a different ESQ voltage, which yielded tunes further away from this vertical low order resonance. The comparison illustrated significant differences in the stability of phase space regions close to the collimators, confirming that the low order resonance noticeably impairs the systemโ€™s long-term stability. The normal form defect analysis was also able to identify the strong ninth order nonlinearities of the map caused by the 20th-pole of the ESQ potential. 243 APPENDIX 244 A.1 Toy Example for Verified Optimization of Four Particles in 3D In literature, the trivial case of four particles in 3D is often discussed as a toy problem, which we run below to provide our results for comparison. In [59], the variable definitions were chosen similarly to our choice in Sec. 6.2.3.8. The only relevant difference is that [59] used the ๐‘ฅ positions as variables with ๐‘ฃ0๐‘ฅ,๐‘– = ๐‘ฅ๐‘–+1 (1) instead of ๐‘ฃ ๐‘ฅ,๐‘– = ๐‘ฅ๐‘–+1 โˆ’ ๐‘ฅ๐‘– as defined in Eq. (6.116). Note that the first particle is fixed at the origin (๐‘ฅ1 = 0) also in [59]. We run the global optimization with Taylor Models of order 5, without providing an initial cutoff value and with a threshold for the smallest boxes of ๐‘ min = 10โˆ’6 , just like in [59]. The initial variable search domains of [59] and initial variable search domains of our optimization are listed in Tab. A.1. Tab. A.1 also shows our results for the optimized variables. Note that we used the same size for the initial search volume as in [59], which is 2.88 ร— 10โˆ’4 . COSY-GO reduced the search domain to 7 remaining boxes with a total volume of 7.5 ร— 10โˆ’41 in 2.102 seconds and 2794 steps. The minimum was bound by [โˆ’3.115103401910087E-307, 8.060219158778647E-14]. (2) Table A.1: The left columns list the variables of [59], denoted by ๐‘ฃ0ยท,๐‘– , and their respective initial search domains. The middle columns list the variables of our optimization and their respective initial search domains. The right columns show the optimized variables of our optimization. Initial domain Initial domain Optimized result ๐‘ฃ0๐‘ฅ,1 [0.4, 0.6] ๐‘ฃ ๐‘ฅ,1 [0.4, 0.6] ๐‘ฃโ˜…๐‘ฅ,1 [0.499999879, 0.500000183] ๐‘ฃ0๐‘ฅ,2 [0.4, 0.6] ๐‘ฃ ๐‘ฅ,2 [0, 0.2] ๐‘ฃโ˜…๐‘ฅ,2 [โˆ’0.445014773E-307, 0.229985454E-6] ๐‘ฃ0๐‘ฅ,3 [0.8, 1.2] ๐‘ฃ ๐‘ฅ,3 [0.4, 0.8] ๐‘ฃโ˜…๐‘ฅ,3 [0.499999879, 0.500000183] ๐‘ฃ0๐‘ฆ,2 [0.7, 1.0] ๐‘ฃ ๐‘ฆ,2 [0.7, 1.0] ๐‘ฃโ˜…๐‘ฆ,2 [0.866025282, 0.866025501] ๐‘ฃ0๐‘ฆ,3 [0.2, 0.4] ๐‘ฃ ๐‘ฆ,3 [0.2, 0.4] ๐‘ฃโ˜…๐‘ฆ,3 [0.288675006, 0.288675372] ๐‘ฃ0๐‘ง,3 [0.7, 1.0] ๐‘ฃ ๐‘ง,3 [0.7, 1.0] ๐‘ฃโ˜…๐‘ง,3 [0.816496487, 0.816496660] 245 BIBLIOGRAPHY 246 BIBLIOGRAPHY [1] Babak Abi et al. (Muon ๐‘”-2 Collaboration). Measurement of the positive muon anomalous magnetic moment to 0.46 ppm. Physical Review Letters, 126:141801, 2021. [2] Tareq Albahri et al. (Muon ๐‘”-2 Collaboration). Beam dynamics corrections to the run-1 measurement of the muon anomalous magnetic moment at Fermilab. Physical Review Accelerators and Beams, 24(4):044002, 2021. 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