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' ‘. . , x ‘l finanemjay .4; I --"'".."W i a. - __ ~ ._ -v..o-D—.-. gm This is to certify that the dissertation entitled BIFURCATION OF PERIODIC ORBITS OF NONPOSITIVE DEFINITE HAMILTONIAN SYSTEMS presented by Yong—In Kim has been accepted towards fulfillment of the requirements for Ph.D. degreein Mathematics {b/kl, (/st Major professor Date November 13, 1986 MS U is an Affirmative Action/Equal Opportunity Institution 0-12771 MSU LIBRARIES ”— RETURNING MATERIALS: lace in book rop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. . 9‘" f- a l- BIFURCATION OF PERIODIC ORBITS OF NONPOSITIVE DEFINITE HAMILTONIAN SYSTEMS By Yong-In Kim A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1986 A”; I» mess» ABSTRACT BIFURCATION OF PERIODIC ORBITS OF NONPOSITIVE DEFINITE HAMILTONIAN SYSTEMS By Yong-In Kim In this thesis, we consider the bifurcations of periodic solutions of a family of non-positive definite Hamiltonian systems of n degrees of freedom near the origin as the family passes through a semisimple resonance. We begin with a smooth Hamiltonian H with a general semisimple quadratic part H2 and then construct a normal form of H with respect to H2 up to fourth order terms and make a versal deformation. We apply the Liapunov-schmidt reduction in the presence of symmetry and further reduce the resulting bifurcation equation to a gradient system. Thus, the study of periodic solutions of the orginal system is reudced to finding critical points of a real—valued function. As an application. we consider a system with two degrees of freedom in 1: -1 semisimple resonance by using suitable choices of the parameters to study the bifurcation as the eigenvalues split along the imaginary axis or across it and we obtain complete bifurcation patterns of periodic orbits on each energy level. To my father and mother who gave me mental heritage. 11 ACKNOWLEDGEMENTS I am indebted to my advisor, Professor Shui—Nee Chow, for his suggestions of the problem and the methodology. consistent support and encouragement, and frequent advice and comments during my work on this dissertation. I have very much benefited from his extensive knowledge and keen insight about the problem as well as his nice personality. Also, I am owing greatly to Professor Richard CUshman for his helpful hints and nice lectures on normal forms during his visit at Michigan State University for one month last year. I am grateful to Miss Sherrie Polomsky for her most efficient and elegant typing of my rough manuscript. TABLE OF CONTENTS Page INTRODUCTION 1 CHAPTER 1. Hamiltonian Systems and Normal Forms §1. Hamiltonian Mechanics 5 §2. Normal Forms for Hamiltonian Functions 18 CHAPTER 2. Versal Deformations of Quadratic Hamiltonians §1. Versal Deformations of Linear Systems ‘ 30 §2. Computation of VerSal Deformation of H2 37 CHAPTER 3. Liapunov-Schmidt Reduction with Symmetry §1. Introduction 43 §2. Liapunov-Schmidt Reduction 47 §3. Reduction to a Gradient System 56 CHAPTER 4. Two Degrees of Freedom 1: —1 Semisimple Resonance Problem §1. Normal Form and Versal Deformation 62 §2. Invariant Manifolds of the Linearized System 66 §3. Eigenvalues of the Perturbed Linear System 73 iv §4. Local Bifurcations of Periodic Orbits as the Eigenvalues Split Along the Imaginary Axis 77 §5. Local Bifurcations of Periodic Orbits as the Eigenvalues Split Across the Imaginary Axis 98 BIBLIOGRAPHY 116 Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure eeeeeeeeee LIST OF FIGURES vi Page 19 70 71 76 88 90 91 93 95 106 108 109 1 1 1 113 INTRODUCTION This thesis is mainly concerned with the study of the bifurcations of periodic solutions of a family of non-positive definite Hamiltonian systems of two degrees of freedom near an equilibrium as the family passes through the 1=—1 semisimple resonance. We start with a smooth (Cm) Hamiltonian function H = H2+ H3+ H4+ with a given normalized quadratic part 0 1 2 2 1 2 2 ( ‘1) H2(z) = § O for all j. there exist at least n distinct periodic orbits on each energy level H(z) = c for 0 < 0 << 1. The essential point in Weinstein’s proof is that the condition of positive definiteness of the Hessian matrix D2H(0) implies the compactness of the energy surface H(z) = c for small c > 0 and so one can apply either a theorem of Krasnoselski or the theory of Lyusternik-Schnirelman to obtain the desired result. If the Hamiltonian is not positive definite, however, then the energy surface H(z) = c.is no longer compact and so the situation is more complicated. Moser [39] presented an example in which D2H(0) = diag (1, -1, 1, -1) and the Hamiltonian system possesses no nontrial periodic solutions. More significantly, Chow and Mallet-Paret [14] proved that if H has the form n +y?) —1- 2 (x§+y§) +0(Izl°). 0.3 H = g ( ) (z) (xJ J 2j=1+1 Rabi II'MH J 1 and is analytic, then the corresponding Hamiltonian system -I z = JV H(z), where J = [O In]. In: n x n identity matrix n 0 actually possesses at least In-le one—parameter families of periodic solutions near the origin provided that there are no 21r-periodic solutions on the zero energy level H(z) = 0. If 8 = n. then H(z) is positive definite and clearly there are no 2w-periodic solutions on the surface H(z) = O and hence this result recovers a part of Weinstein’s theorem. However, if 8 = 3-, e.g., n = 2 and e = 1 (i.e., 1: -1 resonance) then this result doesn’t give any information about the existence of periodic orbits and actually Moser’s example shows the nonexistence of nontrivial periodic solutions. Recently, van der Meer [35] studied the periodic solutions of a family of Hamiltonian systems passing through the 1: -1 nonsemisimple resonance by examining the fibres of the normalized energy-momentum mapping by using the singularity theory of equivariant mappings. In this thesis, we study the same 1: -1 resonance but the semisimple case which has four parameters in a versal deformation of H2 and nine fourth order terms in the normal form in contrast to the non-semisimple case which contains two versal deformation parameters and three fourth order terms in the Hz—normal form. Moreover, our approach to examining the periodic solutions, after normalization. is a local analysis by using the theory of Liapunov—Schmidt reduction in the presence of symmetry and reducing the resulting bifurcation equation to gradient system and studing the critical points of the reduced gradient system. We use the Lagrange multiplier method and take advantage of the equivariance symmetry of the gradiant system to solve it in a closed form. This thesis is organized as follows. In chapter 1. we give a brief outline about the Hamiltonian systems and the theory of Hamiltonian normal forms. In Chapter 2, we introduce the theory of a versal deformation of linear systems and construct a versal deformation of H2 given in (0.3). In Chapter 3, we use the Liapunov-schmidt reduction to examine the periodic orbits of a family of Hamiltonian systems in a normal form and obtain a real—valued function whose critical points correspond to periodic solutions of the original Hamiltonian systems. The summary of our method will be stated in Theorem 3.3.3 as a main theorem of this thesis. Finally, in Chapter 4. we apply our method in Chapter 3 to the 12 — 1 semisimple resonance problem with H2 given by (0.1) under some restriction on the parameters and nonlinear terms and obtain explicit bifurcation results which will be summarized in Theorem 4.4.1 and Theorem 4.5.2. We conclude with a remark about the extension to a nearby nonintegrable systems and other possible methodologies to examine the periodic solutions. CHAPTER 1: HAMILTONIAN SYSTEMS AND NORMAL FORMS In this chapter, we will give a brief review of some basic facts about the Hamiltonian mechanics, and normal forms of Hamiltonian functions which form a background in the following chapters. Even though we are mainly working on the Euclidean space len, the basic structure of Hamiltonian systems will be given in the context of symplectic manifolds since the phase space of a Hamiltonian system is generally a manifold rather than Euchidean space especially when constraints are present. Most definitions and theorem will be stated without proof. For the proofs and more detailed treatments of the above basic theories, we refer to the textbooks of Abraham and Marsden [1] and Arnold [3] and the lecture note of Cushman [22] and the thesis of van der Meer [35]. §1. Hamiltonian mechanics. Let M be a smooth connected manifold. Definition 1.1.1. A symplectic form w on M is a closed, nondegenerate 2—form on M} that is, dw = O and for each m e M; the skew—symmetric bilinear mapping w(m)= TmM x TmM 4> R is nondegenerate (i.e., w(m)(vm,wm) = O for all Wm € TmM Implies vm = O.) The pair (M,w) is called a symplecticgmanifold. Theorem 1.1.2. Let to € 02(M). i.e., a 2-form on M. Then w is nondegenerate iff M is even-dimensional, say 2n. Definition 1.1.3. Let (M,w) be a symplectic manifold and H: M a R a given. cr function. r 2 1. The vector field XH determined by the condition (1.1.1) m(XH,Y) = dH - Y is called the Hamiltonian vector field with Hamiltonian function H. We call (M,w,XH) a Hamiltonian system. We will suppose H to be C00 in the following. Note that the nondegeneracy of w guarantees the existence of XH’ which is a or—1 vector field. Indeed, since (0(m) is nondegenerate, the linear map # , * . # _ w (m)- TmM 4 Tm M defined by w (m)(vm) wm _ w(m)(vm.wm) for all wm € TmM. is invertible. Since dH(m) 6 Tm*M, we have x (m) = w#(m)_1 - dH(m) e T M H m ' Let H(M) be the space of smooth vector fields on M and %*(M) be its dual space, i.e., the space 01(M) of one-form fields on M. For x e a(M) and m e 02(M), define ixw e a*(M) by in(Y) = w(X.Y). We call ixw the inner product of X and w. Then, alternatively, we may define the Hamiltonian vector field by the relation (1.1.2) iXHw = dH. That is, for each m 6 M and each vm € TmM. dH(m) - vm = (ixflwxvm) = w(m)(xH(m).vm). The following theorem shows that the definition 1.1.3 is locally equivalent to the classical one. Theorem 1.1.4. (Darboux). (M,w) is a symplictic manifold iff there is a chart (U,¢) at each m 6 M such that ¢(m) = 0, and with ¢(u) = (x1(u), . xn(u), y1(u), . . . yn(u)), we have The charts (U,¢) guaranteed by Darboux's theorem are called symplectic . . i i . charts and the coordinate functions x ,y are called canonical or symplectic coordinates. Theorem 1.1.5. Let (x1, . . . Xn‘ yi, . . . yn) be canonical coordinates for w, so w = 2 dxi A dyi. Then in these coordinates 6H 6H (1.1.3) XH = ['ay—i, — 'aX—i] - J dH O I ]. Thus, (x(t). y(t)) is an integral curve of XH iff 0 where J = [ -I Hamilton's equation hold: proof iXHm = 2 ixH(dxi dyi) = 2 (iXHdXi) dyi — 2 dxi (iXdei) 6H 6H dH F 2 (6x. dxi + By. dyi) 1 1 we have 6H 6H 1 dx. = -—-a 1 dy. - - XH 1 ayi XH 1 6x1 flutis 6H 6H XH _ (5§;, — 5;?) = J dH. /// Note that if M = R2n, then we have global canonical coordinates (Xi’yi) and w = E dxi A dyi, hence the Hamilton’s equation in (Rzn. w = 2 dxi A dyi) is globally given by (1.3), which is our classical definition of Hamiltonian system. (R2n.w) is called the standard symplectic space. The conservation of energy for the Hamiltonian system is given by the following theorem. Theorem. 1.1.6. Let (M,w,XH) be a Hamiltonian system and 1(t) be an . . d1 t Integral curve for XH' that Is. -a%—L = XH(7(t)). Then H(7(t)) = constant in t. — . - 5‘ '7' (‘3er .n ‘. ' ' .‘Ji... .,.._.,1. _. t .‘ ' The next basic fact about the Hamiltonian systems is that their flows consist of canonical transformaitons. Definition 1.1.7. A Cm— map F2(M,w) % (M,w) is called a symplectic or canonical transformation if F*w = m, where F*: 02(M) a 02(M) is defined by x (F w)(m>(vm.wm) = w(F(m)) - (dp(m). vm. dF(m) - wm) for v ,w E T M. m m m Theorem 1.1.8. Let (M,w,XH) be a Hamiltonian system and ¢t be the local flow of XH. then, for each t, otm = m. that is, ¢t is a local one—parameter group of symplectic diffeomorphisms (on its domain). Thus ¢t also preserves the phase volume Ow (Liouville's theorem). Theorem 1.1.9. If ¢ is a symplectic diffeomorphism of (M,w), then ¢*XH = X¢*H for every H e Cm(M). That is, a symplectic change of coordinates maps a Hamiltonian vector field into a Hamiltonian vector field with Hamiltonian ¢*H. Definition 1.1.10. For X e %(M) and f 6 Cm(M), define fo e c”(M) by (LXf)(m) = df(m)~X(m). We call fo the Lie derivative of f with respect to X. n Note that if H e c“(m2n,m) with w = 2 dxi c dyi. then i=1 10 “ 6H 6 6H 6 (1.1.4) LXH = 2 (~—— -—— - —-— ———). i=1 ayi 6x1 6X1 ayi Now, we introduce the definition of Poisson bracket on Cm(M,R) to impose a Lie algebra structure on Cm(M.R). Definition 1.1.11: Let (M.w) be a symplectic manifold and let f, g 6 Cw(M,R). The Poisson bracket ( , )2 Cm(M) x Cm(M) 4 Cm(M) is defined by (1.1.5) (f,g}(m) = w(m)(Xg(m), Xf(m)) for each m 6 M. Notice that from (1.1.1) and (1.1.4) and (1.1.5), we may write (1.1.6) (f,g} = dg ~ xf = fog = —LX f. g Since w is skew-symmetric, so is ( , ). Thus, in canonical coordinates (Xi’yi)‘ (f,g) may be written as n _ a; 65 _ 6f 65 (1.1.7) {f.s} -i§1(5§;'axi axi 6yi ‘ From (1.1.6), it is clear that f is constant along the orbits of Xg (or g is constant along the orbits of Xf) iff (f,g) = 0. Note that (f,f) = 0 corresponds to conservation. of energy for the Hamiltonian system (M,w,f). We say that F 6 Cm(M,R) is an integral for the system (M.w,H) if (H,F} = o. Definition 1.1.12: A Lie algebra is a vector space V with a bilinear operation [, ] satisfying (i) [X,X] = 0 for all X 6 V and (ii) [X,[Y,Z]] + [Y.[Z.X]] + [Z,[X,Y]] = O (Jacobi identity) for all X,Y,Z. 6 V. Since ( , ): Cm(M) x Cm(M) a Cm(M) is a skew—symmetric bilinear form and satifies Jacobi identity, the real vector space (Cm(M,R), ( , )) together with the Poisson bracket is a Lie algebra. Theorem 1.1.13: ¢ is a symplectic diffeomorphism of (M,w) iff ¢ preserves Poisson brackets, that is, x x x ¢ {f.s} = {¢ f. ¢ g} for all f,g 6 Cm(M.R). Thus. ¢* is a Lie algebra isomorphism on C”(M.m). The next fact is that Hamilton's equation may be written in Poisson bracket form. Theorem 1.1.14: Let XH be a Hamiltonian vector field on a symplectic manifold (M,w) with Hamiltonian H and local flow ot. Then, for every f (X) 6 C (M.R), j—cu ~ .19 = (H. 1 . ,t) = LXHU . t.)- 12 In particular, if f = x1 or yi. we have ’21 = {H’xi} = 3%" (1.1.8) i . -6H 3'1: {H'yi} E So far, we considered the Lie derivative fo for f 6 Cm(M,R). We can also define the Lie derivative LXY for Y 6 H(M). Theorem 1.1.15: If X,Y 6 fl(M), then [LX’LY] = LXI.Y - LYLX is an (R linear) derivation on Cm(M,w). that is. for f,g 6 Cm(M,m), [LX,LY](f°g) = ([LX.LY]f)g + maxing). Definition 1.1.16: For X,Y 6 MM), let [X,Y] = LXY be the unique vector field such that L[X,Y] = [LX,LY]. We call LXY the LIE derivative of Y with respect to X, or the Lie bracket of X and Y. Notice that [ , ] is a skew-symmetric bilinear form on 91(M) and satisfies the Jacobi identity and hence the space of smooth vector fields together with the Lie bracket (H(M), [ , ]) forms a Lie algebra. In the local coordinates, [ , ] is written as (1.1.9) [X,Y] = DY ° X — DX ° Y. The following theorem shows the relationship between the Lie bracket of Hamiltonian vector fields and the Poisson bracket of smooth functions. (X) Theorem 1.1.17- For f,g 6 C (M), [Xf,Xg] = X{f'g}. Thus, the space of Hamiltonian vector fields with Lie bracket (dH(M), [ ]) forms a Lie subalgebra of the Lie algebra of all smooth vector fields on M. The mapping p: Cm(M,R) » dH(M) defined by p(f) = Xf is a homomorphism of Lie algebras (Cm(M), ( , )) and (%H(M),[ , ] ). Definition 1.1.18: For each F 6 Cm(M,R), define the map (X) 00 adF: C (M,R) » C (M.R) by adFG = (F,G}. We call the map ad: Cm(M,R) a L(Cm(M,R), Cm(M,R)): F a adF the adjoint respresentation of Cm(M,R). Notice that for each F 6 Cm(M,R) adF is an inner derivation of Cm(M,R) since, by the Jacobi identity, we have adF{G,H} = (adFG.H} + (G, adFH} for all G.H 6 Cm(M,R). Also, because of (1.1.7), adF has local expression Definition 1.1.19: For H 6 Cw(M.R). the Lie series is defined formally 14 as w n exp ad = E —T-ad H n=0 n. H o . where adH= 1d. adH — adH adH for n 2 1. The Lie series is the essential tool for computing normal forms of Hamiltonian functions. In the followimg some basic facts about lie series are stated. Theorem 1.1.20: Let H 6 Cm(R2n,R) with coordinates (x,y) = (x 1 Q n x ,y , . . . ,y ) and standard symplectic form w = 2 dx. A dy.. Then, n 1 n i=1 1 1 (i) adH(x,y) = XH(x.y), where adH(x,y) 2 (adel, . . . ,aden, adHyl, . . . , adHyn). (ii) exp(t adH)°(x,y) is the flow of XH' (111) For any r e c”(m2n,m), (F . exp adH)(x,y) = (exp (adH) . F)(x.y). (iv) exp adH and exp ad commute iff {H.F} is constant iff [XH,XF] = O. F Notice that the space {adFl F 6 Coo(ll22n,lR)} is a Lie algebra with bracket [adF. adG] = ad or [XF,XG] = X(F,G) If we identify the {F.G} vector field X with its Lie derivative LX. Hence, the set C = {exp adFI F 6 Cm(R2n,R)} forms a Lie group. Then, each one-parameter group {exp t adF: t 6 IR} forms a one-parameter subgroup of G. On the 15 symplectic space (Rzn,w) each one-parameter group of symplectic diffeomorphisms is the flow of a Hamiltonian vector field. Thus, we have found all one-parameter subgroups of G because each generator of C is a symplectic diffeomorphism which is the time one flow of a Hamiltonian vector field. Definition 1.1.21: Let (RZn.w) be a symplectic space. A linear map ¢:m2n sing“ is symplectic iff w(¢v, ¢w) = w(v,w) for each v,w 6 R2n. The set of all linear symplectic mappings of (Rznnw) is a Lie group Sp(n,R) called the real gymplectic group. A linear map A: R2n A>R2n is infinitesimallv symplectic iff w(Av,w) + w(v,Aw) = 0 for every v,w 6 R2n. The set of all infinitesmally symplectic maps is a Lie algebra sp(n,R) under the Lie bracket [A,B] = BA - AB. Note that A 6 sp(n,R) iff eA 6 Sp(n,R), which relates the Lie algebra to the corresponding Lie group. Theorem 1.1.22: Let ¢ 6 Sp(n,R) and A 6 C be an eigenvalue of ¢ of multiplicity k. Then -1-, A, :1-_are eigenvalues of d: (A = complex A A conjugate of A) of the same multiplicity. Theorem 1.1.23: Let A 6 sp (n,R) and A 6 C be an eigenvalue of A of multiplicity k. Then, —A, A: -A are eigenvalues of A with the same multiplicity. . , 2n , 2n 2n . Definition 1.1.24- On (IR .09), the map (I) G x [R -9 [R 15 called a symplecticAaction of the Lie group G on m2n if for each ¢ 6 G, the map 16 (RP : IR2n #112211: x 4¢(¢.X) is symplectic. . 2n . . In a natural way, the action (D on IR induces an action of G on c°°(lR2n,IR) \II: c x c°°(1R2n,1R) —> c°°(m2n,1R): (¢,H) —» H - (1)45. we often write ¢ ° H for @(¢.H). Definition 1.1.25: 1A Lie group G acting symplectically on R2n is a symmetry group for the system (Rzn,w,H) if ¢ ° H = H for all ¢ 6 G. Theorem,1.1.26: If I: is an integral for the system (R2H,w,H) i.e., (F,H} = 0, then the one-parameter group {exp (t adF): t 6 R} given by the flow of X is a symmetry group for (Rzn,w,H). F! The converse of the above theorem also holds in the sense that each symmetry group of a Hamiltonian system gives rise to an integral. To make this precise, we first introduce the notion of momentum mapping. Definition 1.1.27: Let ¢>be a symplectic action of the Lie group G on (Rzn,w) with the Lie algebra L. The mapping J: m2n 4»L* is a momentum mapping for the action ¢>if for every § 6 L 17 d x,. (X) = — ¢(exp t 5.x} _ where the right-hand side is called the infinitesiml generator of the Zn action corresponding to f and J(§) 6 Coo(lR ,IR) is defined by 3(§)(x> = J(x) . 5. Theorem 1.1.28: Let (I) be a symplectic action of the Lie group G on (lenxu) with the momentum mapping J. If G is a symmetry group for (R2n,w,H), then {J(§),H} = O, i.e. 3%) is an integral for (R2n,w,H). 18 §2. Normal fortfifor H_a_miltonign functions In this section, we will assume that H 6 Coo(IR2n,IR) with H(O) = O and dH(O) = 0, that is, the origin 0 of R2n is an equilibrium point for XH' The goal of normal form theory is to find an origin-preserving symplectic diffeomorphism ¢ of R2n which preserves the Hamiltonian character such that H in the new coordinates defined by ¢. i.e. , ¢*H = H °¢ is in the simplest possible form. 2n Let 9: (IR 2n ,IR) be the space of all formal power series on IR beginning with terms of degree 7 2 2, and 9PJ.(IR2n,IR) be the space of homogeneous polynomials on IR211 of degree j. Let G be the Lie group of all origin-preserving symplectic diffeomorphisms on IR2n of the form id + 45(2) where ¢(2) is an IRzn-valued formal power series all of whose components lie in 932+(IR2nJR). The action (I) of G on IR2n induces an action ' on 9P:(IR2n,IR) given by ‘1’ ° F = ¢(¢,F) = ¢*F for 4’ 6 G. F 6 952+(IRZDJR). Let QG(H) = {95 ° HI ¢ 6 G} be the orbit of H under the action of G. Let C(Hz) be a complementary space to QG(H2) at Hz. Definition 1.2.1. Let H. H 6 932+(IRZHJR) and H = H2 + H3 + with H]. 6 93(IR2DJR). Then we say H is a H,—norm_al form for H if H 6 QG(H) fl C(Hz). Note that the tangent space to QG(H2) at Hz is H2 + adH 9153+ because 2 for all F 6 953+, t -> exp t ad is a one-parameter subgroup of G which F represents the tangent vector X to G at id. Therefore, t -> (exp t F 19 ad?) . H3 13 a curve in Q(H,) passing through H2. Thus. the set of tangent vectors d d 2 - 2 E- t==0(exptadF) °H -d—t-|t=oexp(tadF)H ad FHZ - adeF for all F 6 93+(R2n,R) is TH Q(H2). 2 If Cm is a complementary subspace to the image of the linear map adHZI 91m: 9m figm: f -’ {H2,f) for all m 2 3. then the subspace 0° + 2 + C = 2 C of 93 (R n,R) is complementary to Im adH of 973 . See Figure m 2 m=3 1.2.1. __. C‘I'H, II t m “AH P3++Hz fl: \ 7- Cl(Hz) < Figure 1.2.1 > The following theorem gives an algorithm for finding a formal power series symplectic diffeomorphism ‘I’F which brings the formal power 20 series Hamiltonian H = H2 + H3 + ... into normal form. The normalizing transformation ¢F is constructed by induction. Theorem 1.2.2. Let H e 92+(m2nm) and let H2 as o be the quadratic part of H. Then, for each m 6 N, m 2 3, there exists a ¢ 6 G such that H'z H ° ¢ is in the H2- normal form for H up to order m. (proof) Suppose that H'is in H2 - normal form up to terms of degree m-1 2 2, that is, suppose there is a F(m_1) 6 93+ such that Hm): ~H=H2+H3+...+H +H+... where H; = H2 if m = 3 and H; 6 Ci for i = 3,4,5, ... , m-1. For m 2 3, let Fm 6 9m(R2n,R). Then we get II .5“ + SKI + + I” + Sm + 8. Therefore, the terms of degree m in the above are Hm + adF H2 = Hm - ad F . m H2 m Since 9 (Rzn,R) = C 9 Im ad I R , we may write H = H'+ H m m H2 m m m m 21 where H. 6 C and H 6 im adH I 9 . Since H 6 im ad I 9 , we can m m m 2 m m H2 m choose a F 6 9 such that adH F = H . With this choice of F , m m 2"! m m m+1 * m -— ._ ._ H( )=¢FmH_()=H2+H3+ +Hm_1+Hm+O(m+1), that is, ¢F brings H(m) into H2 - normal form up to order m. Thus we m have fiflm+1) : ¢ x ¢x H = ¢ . ¢ * H = ¢x H. Fm( F(m—i) I I F(m-I) Fm) F(m) This completes the inductive step of the normalization process. Repeating this step degree by degree gives a formal symplectic diffeomorphism ¢ = ¢F . 3 ¢F ° 4 ’¢F°...:¢F m which brings H into normal form, that is, ¢;.H=Hz+fia+...+fim 9 where H; 6 Cm for all m 2 3. /// + Since the normal form of H = H2 + H3 + ... + H + ... 6 92 depends on m the choice of complement Cm to the image of the linear map adH I 9m: Rm 2 -29;!for all m 2 3, we need to know how to compute Cm in general case. 22 We need some basic facts about linear maps from linear algebra. (See Humphreys [29] , see also CUshman [22].) Definition 1.2.3. Let V be a finite dimensional real vector space. A linear mapping S: V1» V is semisimple if every S—invariant subspace U of V has an S-invariant complementary subspace W of V. S being semisimple is equivalent to saying that S is diagonalizable on the complexification of V. A linear mapping N I V A»V is nilpotent if there is an m 6 R such that Nm = 0 but Nm-1 ¢ 0. Theorem 1.2.4. Let A I V -> V be a linear mapping. Then there are unique simisimple and nilpotent linear maps S and N on V such that SN = NS and A = S + N. The maps S,N given above are called the S-N decomp0§ition of A. The following theorem is very useful in finding Cm in the normal form. Theorem 1.2.5. Suppose A = S + N is a S-N decomposition of a linear mapping A:V etV. Then (a) V = ker S 0 im S (b) ker A = ker S n ker N (c) im A = im S 0 (im N n ker S). The following shows that Theorem 1.2.4 also holds in sp(V,R). Theorem 1.2.6. Suppose that A: (v,w) -> (v,w) is infinitesimally symplectic and has an S - N decomposition A = S + N. Then S and N are also infinitesimally symplectic. Now, the S - N decomposition A = S + N in sp(V,R) propagates into the space (92, { , }) and (gl(@m,R), [ , ]). That is, using the Theorem 1.2.6 and the isomorphism p 1 (92,( , ) )-+ (sp(V,R), [ , ])3 f -> Xf of Lie algebras, every H2 6 92 has a corresponding S - N decomposition H2 = S2 + N2 with (SZ,N2} = O and Sz.N2 6 92. Further, the map ad(m): @2 a’g1(9m,R): f 4'ade 9m = LXfl ?m is a representation of Lie algebra (@2, ( , }) into the Lie algebra (gl(@m,R), [ , ] ) and hence if H2 = S2 + N2 is the S - N decomposition of H2 6 @2, then adH I 2 2 = ad I R + ad I R is the S - N decomposition of ad 9 . From m S2 m N2 m m 112' this fact and Theorem 1.2.5, we have the following important criterion for computing a Hz- normal form for H. Theorem 1.2.7: Let H = H2 + H3 + ... + H + ... 6 G: (Rzn,R) and let m H2 = $2 + N2 be the S — N decomposition of H2. Then, H is in norm form (m) n with respect to H, iff Hm 6 Cm where Cm is a complement to (im adN 2 ker ad (m) ) in ker ad (m) for every m > 3, where ado“) = ad I 9? 82 S N2 N2 2 T m etc. proof By the definition of Cm’ and Theorem 1.2.5(C), we have a = im ad(m)$ c m H2 m 24 _ ~ (m) - (m) (m) — 1m adsz 6 (im asz n ker adS2 ) 6 Cm' Since ad (m) is semisimple. 9 = im ad (m) 6 ker ad (m). Hence, ker ad (m) = (im ad (m) n ker ad (m)) 6 C . Thus, C is a complement of (im ad (m) D ker ad (m)) in ker ad (m). /// N2 Sz 82 Notice that if H2 is semisimple. i.e., H2 = S2 and N2 = 0, then we may take Cm = ker ads(m). Hence, if H2 consists of only semisimple 2 part S2, then we can say that the formal power series H = S2 + H8 + ... + Hm + ... is in S2 - normal form iff Hm 6 ker ad I 6m for all m 2 3. 2 S . w 2n . Recall that we can bring an H = H2 + ... + Hm + ... 6 C2(R ,R) into 82- normal form by a symplectic diffeomorphism ¢F = exp adF ° exp adF a 4 ° ... exp adF ..., where Fn 6 @n(n 2 3). At each step, exp adF is n n determined up to terms of F in ker ad(n)(i.e., F = F' + F , Fl 6 ker n S2 n n n n ad(n), F 6 im ad(n) and F. may be arbitrary.). Starting with n = 3. 82 n 82 n this freedom of choice of F; may lead to different S2 - normal forms up to order > 3. However, these 82- normal forms can be transformed into one another by a symplectic diffeomorphism. Thus, the S2— normal form is essentially unique. Let d(S2) = ker ad I 92+. Then, since d(Sz) is closed under ° 32 and { , } and since ad is a derivation of (62+,°), (d(Sz),°, { , I) 32 is a Poisson structure, that is, (M(Sz), ° ) is an associative algebra with unit over R, while (94(82), { , I) is a Lie algebra. We call (d(S2), ° ) the Birkoff algebra of $2. The main goal of the semisimple case of normal form theory is to describe the Birkoff algebra. because then we know what power series appear in the normal form. The only known general fact about the Birkoff algebra d(Sz) is the following. Theorem 1.2.8. If the semisimple Hamiltonian vector field XS 2 corresponding to 82 has pure imaginary eigenvalues. then d(SZ) is finitely generated. (For the proof, see Cuchman [22]). Now, in order to determine the Birkoff algebra d(Sz) = ker adS 2 for a specific semisimple quadratic Hamiltonian 82 where X has purely 2 S imaginary eigenvalues, we will need to know a normal form for X8 on a 2 symplectic vector space (V,w). Theorem 1.2.9 (Cushman [22]). Suppose H2 = 32 and X8 has pure 2 imaginary eigenvalues :iaj. Then there is a basis (e1, ... ,e , f . fn} of (v,w) such that the matrix of (it: V —» v" defined by w#(e) ‘ e' = w(e,e') is Iw(fi,ej) I w(fi,f.)I = o Id(ei,ej) Iw(ei,ij [0 an] J n and n 1 Sz(x,y) = 2 E ejaj(x§ + y?), i=1 where ej = t 1, aj > O and iejaj are the eigenvalues of XS . Thus the 2 matrix of XS with respect to the above basis is 2 "—2 26 e1a1 O 6.qu x82 6 n _eia1 -eza2 '. 0 _enan /// Now, assuming that X has pure imaginary eigenvalues : ia‘j and $2 is in the normal form given in Theorem 1.2.9., we can compute the Birkoff algebra ker adsz as follows. Let (x,y) be coordinates on R2n corresponding to the basis given in Theorem 1.2.9. Introduce complex conjugate coordinates zj = x3. + iejyj, z = Xj — itajy.j for j = 1, ... ,n when ej are those given in the normal form of XS . 2 Then the linear operator “ a a = ad = 2 e.a.(y.-—— - X.—-— LXS2 Sz i=1 J J 36XJ Jayj n - = ad~ = -i E a.(z ——— - ;;——) Lxs2 SZ i=1 3 62 N n _ N where S = l 2 e.a. 2.2. and X~ is the complex vector field 2 2 :1 J J J J 32 j 27 . = -2i .21 = — i e.a.z. J 62_ JJJ . 21 —2-=ie.a.-z-. forj=1. J azj J J J N| u 2n . . jk J1 The space 9m(R ,R) is the real space of the monomials x y : x1 jn k1 kn xn y1 . . . yn . In complex conjugate coordinates 9m corresponds to the space of Hermitian polynomials 9m(62n, C) which is the . . j _k _k Hermitian span of the monomials 232k = 2131 . . . 2n n 211 . . . znn . N — '—k . — that is, P (2,2) = E C. sz 6 9 if and only if C. = Ck" m . k k |J|+|k|=m J m J J . ~ . . j-k Applying the operator adS to the nomomial baSIS z , 2 ad~ ' szk = - i < j - k, a > szk 32 . . . n where J _ (31, ... , 3n), k _ (k1, ... kn) 6 2+, a _ (a1, . . . an) 6 Rn+ and < , > is the inner product on Rzn with norm I I. Therefore, (X) .-1< kerad§=(2 2 cszz I=OandC.k= 2 m=2 IJI+IkI=m J J C...»- The relation < j - k, a > = 0 is called the resonance relation corresponding to XE . 2 28 The corresponding space of real formal power series in C:(R2n,R) is ker adsz. The normal form theory in the case of H2 = N2 where N2 is a nilpotent quadratic polynomial on (Rzn, E dx.j A dyj) is a little more complicated. By the theorem of Jacobson-Morosov, we may embed N2 into a subalgebra of (92(R2n,R), ( , I) which is isomorphic to sB(2,R), that is, there are M2, T2 6 62 (R2n,R) such that {T2,N2) = 2N2, (T2,M2) = -2M2, (N2,M2) = T2. Then the finite dimensional representation ad”): (ad . i) » (teams). I . ])= f eadfo“) of Lie algebras restricts to a finite dimensional representation of 38(2,R), that is, we have the corresponding commutation reelations, [adT 0“, adN ("0] = 2 adN (m), [adT (m), adM (”)3 = —2 adM (m). (m), adM (m)] = adT (m) From the representation theory of 58(2,R), we have [adN2 ker ad (m) 6 im ad (m) = 6 M2 N2 m for every m 2 3. Therefore, we can say that the formal power series Hamiltonian H = N2 + H3 + . . . + Hm + . . . is in N2 — normal form iff H 6 ker ad (m) 0 ker ad(m)for all m 2 3. m M2 82 In analogy with the Birkoff algebra in the semisimple case, we call the algebra W(N2) = (ker adM , ~ ) the top weight algebra of N2. 2 Also, as a first step in constructing an explicit embedding of N2 into 29 $8 (2,R) we need a normal form for nilpotent infinitesimally symplectic linear mapping XN . Since in this thesis we need only the $2 - normal 2 form, we omit the further details about the N2 - normal form theory. (See Notes on Normal form theory, Richard Cushman 1985, Normal Forms and Symmetry, Sanders 1985). 30 CHAPTER 2: VERSAL DEFORMATIONS 0F QUADRATIC HAMILTONIANS To construct a versal deformation of H2, we need to know a versal deformation of XH in sp (n,R). In Section 1, we will treat 2 briefly the theory of versal deformation of linear systems and in Section 2, we construct a versal deformation of H2 given in (0.3). §1 Verggl deformgtions of Linear Systems The reduction of a matrix in ge(n,R) to its Jordan normal form or a matrix in sp(n,R) to its normal form is an unstable process since both the normal forms themselves and conjugating transformations depend discontinuously on the elements of the original matrices. In this section we introduce the theory of versal deformtions for finding the simplest possible normal form (so called miniversal deformation) to which not only one specific matrix, but an arbitrary family of matrices close to it can be reduced by means of a mapping smoothly depending on the parameters. For further details see Arnold [2][5] and Kocak [30]. Let L be a real Lie algebra with its corresponding Lie group G, e.g.,L may be ge(n,R) or sp (n.R) with G = GL(n.R) or Sp(n,R). Let Ao 31 6 L and Ak be a small neighborhood of the origin of Rk for some integer k. Definition 2.1.1 A deformation A(A) of A0 is a smooth mapping A: Ak'e L such that A(O) = A0. A deformation is also called a family. the variables Ai parameters and the parameter space A = (A) a base of the family. Similarly, we can define a deformtion of an element of G. Definition 2.1.2 Two deformations A(A) and B(A) of A0 are called equivalent if there exists a deformtion C(A) of the identity e of G with the same base such that A(A) = C(A) B(A) c‘1(x). C(O) = e. , 8 k . . . Let ¢: A a’A be a smooth mapping With ¢(0) = O. The mapping ¢ of the parameter space A2 = {u} into the base of the deformation A(A) defines a new deformation (¢*A)(u) of A0 by composition. (¢*A)(u> = A(¢(u))- The deformation ¢*A is said to be induced by A(A) under the mapping ¢. Definition 2.1.3. A deformation A(A) of A0 is called versal if every other deformation B(u) of A0 is equivalent to a deformation induced by A(A) under a suitable change of parameters, i.e., if there exist C(u) and 6 such that 32 B(u) = C(u)A(¢(u))C_1(u) with 0(0) = e. ¢(0) = o. A versal deformation A(A) is called universal if the inducing mapping ¢ is_determined uniquely by B(u). A versal deformation is said to be a miniversal if the dimension of the parameter space A = {A} is the smallest possible for a versal deformation. These miniversal deformations are normal forms with the smallest possible number of parameters in the reduction to which the smooth dependence on the parameters can be preserved. Now, in the following we introduce the important fact that a versal deformation A(A) of A0 is the mapping A transversal to the orbit OfA atA=O. 0 Let Q be a smooth submanifold of a manifold L. Consider a smooth mapping A: A e»L of another manifold A into L, and let A be a point in A such that A(A) 6 Q. Definition 2.1.4. The mapping A: A eiL is called transversal to Q_§t A if the tangent space to L at A(A) is the vector space sum of the image of tangent space to A at A under Ax and the tangent space to Q at A(A), i.e., L = A*T A + T TA(A) A A(A)Q’ where Ax: T A -> T A A(AIL is the push-forward of the map A. 33 Now, consider a Lie algebra L with the corresponding Lie group G. The Lie group G acts on L by conjugation, called the adjoint action as follows. Adge = g2 g‘1(g e c. e e L). The orbit Q(Ao) of a fixed element A0 6 L under the action of G is a smooth submanifold of L defined by -1 Q(Ao) _ {Adeo - ngg I g 6 G}. Theorem 2.1.5 A deformation A(A) of A0 is versal if and only if the mapping A is transversal to the orbit of A0 at A = O. For the proof of this theorem, see Arnold [2]. Our next problem is to determine the mimimum number of parameters for any versal deformation of A0 6 L. From Theorem 2.1.5 we know that in a versal deformation of AD the number of parameters is minimal when the vector space sum in the Definition 2.1.4 is a direct sum. Consequently, this minimum number is equal to the codimension of the orbit of A0 in L. The next argument shows that the direct sum complement of the tangent space of the orbit of A0 is the centralizer of A0 in L if ad is a semisimple linear mapping of L. Let us A o elaborate on this. 34 Let A0 6 L where L is a Lie algebra with its Lie bracket [ ]. Definition 2.1.6. The mapping adA : L -> L is the endomorphism of L 0 defined by adA X = [X,Ao] for all X 6 L. o The kernel of this endomorphism, ker adA = {X 6Z1. I [X,AO] = 0}, is 0 called the centralizer of A0 in L. Theorem42.1.7. The tangent space TA Q(Ao) of the orbit of A0 at A0 6 L o is equal to ImadA0 in L. proof. Consier the mapping for a fixed A0 6 L, . . -1 M%wnuhgeg§ . The image of AdA is the orbit of A0 in L under the action of G. 0 Note that the derivative of AdA at the identity element e 6 G is the o A L. defined by linear mapping (AdA )*: TeG 4'T o o (AdA )*o x 0 II l—l X > O |__1 35 = ad ° X for all X 6 L. Since T80 and TA L are isomorphic to the Lie algebra L, the above 0 calculation shows that (AdA )* = adA . O 0 = Im adA . /// Therefore, TA Q(Ao) = Im (AdA o )x- 0 0 Notice that since adA is an endomorphism of L, o dim L = dim (Im adA ) + dim (Ker adA ). o 0 Hence, the dimension of the centralizer of A0 is equal to the codimension of the orbit of A0 in L. Thus, the problem of constructing a miniversal deformaiton of A0 is reduced to finding a direct sum complement to Im adA in L. 0 Now, let us consider the problem of finding a versal deformation of H2 in the space 92(R2n,R) of homogeneous quadratic polynomials on R2n. We already know that the Lie algebra (sp(n,R), [ , ]) is isomorphic to the Lie algebra (92, { , }) by the isomorphism p: H2 6 XHZ. Hence, if A0 = XH2 6 sp (n,R) and has a S - N decomposition, then H2 has a corresponding S - N decomposition H2 = $2 + N2 with {32.N2} = O and 82, N2 6 92. Furthermore, since the map ad(2): 92 e>g£(92,R): H2 36 -9 ad I = I 62 is a representation of Lie algebra (92. { . }) H2 92 2 into. the Lie algebra (g£(92.R), [ , ]). ad 92 also has a H2 I corresponding S - N decomposition ad I 92 = ad I 92 + ad I 92. H2 SZ N2 Let Sp(2,R) act on 92 by composition. Then, the tangent space to the orbit of H2 at H2 is given by Im ad Thus, to construct a Hz' miniversal deformation of H2 we have to determine a direct sum complement to Im ad If H2 = $2 + N2, then from the Theorem 1.2.5(c) Hz' and Theorem 1.2.7 such a complement C2 is given by the complement of (Im ad (2) n ker ad (2)) in ker ad (2). In particular , if H2 = S2 N2 $2 82 (N2 = 0) then C2 = ker ad (2) in 92. The ker ad (2) is just a Lie 2 32 s subalgebra of 92 isomorphic to the centralizer of X i.e., ker ad X 82 32 in Sp(2,R). If H2 = S2 + N2 (N2 ¢ 0), then to find the complement of Im adN 2 in ker adS we embed N2 into a subalgebra of (92, { , j} ) which is 2 isomorphic to 58(2,R) and is spanned by N2, M2, T2 6 9132 with the commutation relations: {T2.N2} = 2N2, {T2,M2) = - 2 M2, and (N2,M2} = T2. Then the finite dimensional representation ad‘2’= (ere. { . n a (await). [ . 1): H, —>ad 2(2) H of Lie algebras restricts to a finite dimensional representation of sB(2,R) and has the corresponding commutation relations. From the representation theory of sé(2,R), we have 37 62 = ker adM (2) 6 im adM (2). 2 2 Hence, the complement C2 of im adN (2) in ker adS (2) is given 2 2 by C2 = ker adM (2) fl ker adS (2). 2 2 Since in this thesis we only consider the case H2 = 82 (N2 = O) we omit the further details for the nonsemisimple case. See CUShman [18,19,20,21,22] and van der Meer [34,35]. §2. Computation of Versal deformation of H9 Consider a Hamiltonian H 6 C00 (Rzn, R) with H(O) = dH(O) = 0, H(X,y) = H2(XvY) + H3(X’Y) + H4(X’Y) + . 2n where z _ (x,y) _ (x1, . . ., Xn’ yl, . . . yn) 6 R Hj(z) = a homogeneous polynomial in x,y of degree j, and H2(x,y) is the nonpositive definite quadratic form given by (b n l l g + 6 - - E x? + 3 = -z Az (XJ 3’3) 2 j=e+1( J yJ) 2 1 (2.2.1) H2(x,y) = §-.E 3-1 where A is 2n x 2n real diagonal matrix of the form 38 (2.2.2) A = diag (1, ... 1. -1, .... -1, 1, .... 1. -1, ..., -1). Now. we try to find a versal deformation of H2 given above. For each H 6 Coo (RszR). consider the adjoint map adH: Cw(R2n,R) -) Cm(IR2n.R) defined by n 6H 6 6H a (2.2.3) ad = 2 (——-- ——) = {H.°} H . . 6 . 6 . 6 . J=1 ayJ xJ xJ yJ where ( , } is the Poisson bracket on the Lie algebra (Cm(R2n, R), { . )). Then from (2.2.1), we have 2 n 6 6 6 6 (2.2.4) ad = 2 (y. ———-— x. ——— ‘- 2 (y.———-— x.-—-. Lt.=.+°..—.=.-'.. '=l,..., . e zJ xJ 1yJ zJ xJ 1yJ (j n) In complex conjugate coordinates (2,2) = (21, . . ., 2n, 2}, 2'), we have n N __ 1 2 __ 1 n H2(x,y) = H2(z,z) = - 2 z z — - 2 z e n (2.2.5) ad’fi (2.2) a -i 2 (2.5g— - 2%) — >3 (2:52— - ~97) 2 i=1 3 j Jaz j=8+1 J J Jaz Let @m(z,2]= the space of real homogeneous polynomials in 2,2 of degree m. Let Pm(z,z) 6 @m(z,z). Then we may write 39 _ arfi ._ P (z,z) = 2 C 2 Z . C = C , m |a|+IB|=m°43 “‘3 5“ G15 a1 a2 an_p1 _pn where z z = z z . . . z z . . . z and 1 2 n 1 n Ial + '6' = (a1+ . . . + an) + (51+ . . . + fin), aj. Bj = nonnegative integers (jzl, . . ., n). N Applying the operator adH to the basis monomial zagfi. we have 2 n (“3‘53" 2 (aj-BJ.) (za'z'fi). 1 jze+1 Hence. zaifi € ker ad~ if and only if H2 @ ' 2 n a 2 a.- . - 2 a.- . = O and () (3:33) (JBJ) (2.2.6) 4 3‘1 j=e+1 ((b) lal + IBI = m. Now. we compute the Hilbert generators zaE—B for the Birkoff algebra ker adfi 9 in order to construct the versal deformation of H2(x.y). 2 2 Now. the conditions (2.2.6)(a). (b) can be rewritten as P e n 8 n 2 a. + E B. = 2 B. + E a. i=1 J j=e+1 J j=1 J j=e+1 J 2 n 8 n 2 a. + E a.+ E B. + 2 5.: m . :1 J j=£+1 J j=1 J j:e+1 J .J' 40 Let a' = (a1. . . ., a3), a" = (ae+1. . . . .. an) B' = (Bl. . . .. Be). B" = (53.1. . . . .. 5n) Ia'l = a1+ . . . + ae. etc. Then, the conditions (2.2.6)(a). (b) can be written as { (a) Ia'l + IB"I = IB'I + la"| (2.2.7) (b) la'l + Ia"l + IB'I + IB"| = m. Now, For m = 2. (2.2.7) implies la'l + IB"| = IB‘I + la"| = 1. Hence there are 4 possible solutions for Ia'l, Ia"l, IB'I, IB"II rIa'l Ia"| IB'I |B"| 1 1 O 0 (2.2.8) fl 1 O 1 O O 1 O 1 O O 1 1. Let 2' = (21, . . .. 29). z" = (ze+1 . . ... zn). Then the Hilbert generators 2025 = (z')a (2..)a (2")3 (53.)5 for ker ad take the form: H2 22 (2-2-9) (z')“'(z")“". (z )a (E )B'. (2") "(E ')B" (2")5'<2"')B" where Ia'l = Ia"! = IB'I = IB"| = 1 and 41 (z')a (z")a contains e'(n-2) terms 2 1 ze+1' z1ze+2' ' ' " zezn' (z')d.(E")B. contains 82 terms 21 El. 212;. . . .. 2822' (2")a'.(;"')B.. contains (n—e)2 terms 28+1§é+1. 28+IE£+2. . . .zn_£. (2")B.(E"')B.' contains B(n-B) terms E1 22+1. E1E2+2' . EQEQ' Note that each term in (2“)5 (En')B is the conjugate of each term in (z')a (2")a' . Hence. the real Hilbert generators for ker adH are 2 of the form Re (zizj). Im (zizj) (i = 1. . . .. 2, j=8+1. . . ..n) (2.2.10) Re (213?), Im (213k) (i. k = 1, . . .. 8) Re (zizk). Im (zizk) (1. k = 8+1. , n) ._ 1 e ._ Note that each. term: of our quadratic form) H2(z,z) = 2 .2 zJ.z.j - i=1 1 n -— 2 2 z.z. is in the above list. j=e+1 J J Now.let A 8 n (2.2.11) H2(x,y) = i? 2 [aij Re (zizj) + bijIm (zizj)] 1 j=£+1 8 E [C.. Re(zizj) + dijIm (Zizj)] n n + 2 2 [e.. Re (z.21) + f.. Im (z.§k)] i=e+1 j=e+1 ‘3 1 J ‘3 1 J 1 T -511 B(A)“ , 2n where u = (x1. . . .. Xn’ yl. . . ..yn) 6 R . and B(A) = (Bij(k)) 6 R2n x 2n, Bij(x) = coefficient of xiyj. Then. the versal deformation of H2(x.y) is given by A (2.2.12) H2(x,y) + H2(X,y). For m = 2k + 1 (k = 1, 2. . . .) . (2.2.7) implies . .. _ . .. _ m _ 2k + 1 |a|+|l3 |—|B|+|a |—2——2. Since Ia'l + Ia"l = '3'] + [5"] are nonnegative integers, there are no solutions a'. a". B', B" satisfying the above relation. Hence. for m = 2k + 1 (k = 1. 2, . . .), ker ade P = o m i.e., there are no third or higher odd-powered terms in the normal for H with respect to H2. 43 CHAPTER 3. LIAPUNOV - SCHMIDT REDUCTION WITH SYMMETRY §1. Introduction Suppose our Hamiltonian function H(x,y) is in normal form up to a finite order m (=even integer) with respect to H2(X.y) and consider the truncated Hamiltonian function H(x.y) up to order m, (3.1.1) mm) = H.(x.y) + N(x.y). z = (m) 6 m2“. where . 1 e 1 n 1 t H2(x.y)=— 2(x +y)—— 2 (x +y)=—2 A2. 2 . 2 . 2 3:1 J=e+1 A = diag (1, . .. 1. -1. . .. —1, 1. . . 1, -1. . ., -1)eIR2HX2n (3.1.2)< N(X.y) = H4(X.y) + H6(X-Y) + - - + Hm(x.Y)- , 6 k d f k = 4,6 . . .. . (Hk(x y ) er a H2 or m . . . 4A - , Con51der a linear versal deformation H (x,y) of H(x.y)- Aék A (3.1.3) H (x,y) : H2(X,y) + H2(X,y) + N(x.y). Remark: Since HA 6 ker adH , (H2, HR} = O. i.e. H2 and H?‘ are two 2 n integrals for the Hamiltonian system (R2n, w, HA). where m = 2 dxi A i=1 44 dyi is the standard symplectic forms in m2n. Hence. in the case of 2 degrees of freedom i.e..'n = 2. the system (R4. w. HA) is completely integrable with integrals H2 and HR for each value of A. However. the full nontruncated system (R4. w. Hx) where Hx(x.y) = Hk(x.y) + Hm+1 + Hm+2 + . . . with Hj 6 ker adefor j 2 m+1 is not integrable. But. according to the Moser-Weinstein reduction (See [24]). for [AI sufficiently small. there is a Cm function Ex(x,y) depending smoothly on A for O < IzI << 1 such that EA is in Hz- normal and coincides with H up to order m. Hence, the search for periodic solutions of a family of nonrintegrable systems (R‘,w.HK) can be reduced to the search for periodic solutions of a nearby family of integrable systems (R‘.w,Ex). . . 2n Hence. we may restrict our attention to the truncated system (R , m. H—A). From now on. we write H(x.y) for H(x.y) for the notational simplicity. Returning to (3.1.3), consider the .Hamilton’s equation. with Hamiltonian Hx(z) = H2(z) + H§(z) + N(z). z = (x,y) 6 m2n. (3.1.4) 2 Jka(z) JvH2(z) + JvH§(z) + JVN(z) JAz + JB(h)z+ JVN(z). where H2(z) : zTAz. H:(z) = 2TB(h)z. B(A) is given in (2.2.11) with B(O) = o. and 45 (3.1.5) J H H H n x n identity matrix. I l #4 Note that (JA)2 = JAJA = JzA2 _ (JA)2 + I = 0. Hence, JA has eigenvalues i with multiplicity 22. (-i) with multiplicty 2(n—8). Hence. the linearized system of (3.1.4) at z = O. that is. (3.1.6) 2 = JAz + JB(A)z passes through the 1i1i...212 -1=-1t...=-1 resonance when A = 0. At A = 0. (3.1.6) becomes No (3.1.7) = JAZ and (3.1.7) has the solution _ JAt z(t) _ e zo 2n with the initial vector 206 R where JAt e I + JAt + %T(JAt)2 + %T(JAt)3 + . I + JAt — l—-112 - 1 JAt3 + . 2! 3T 1 1 3 I(1 — §Tt2 + ) + JA(t — 37¢ + . . .) H H (cos t) + JA (sin t). 46 That is. (3.1.8) eJAt = I(cos t) + JA(sin t). Therefore. the linearized equation (3.1.6) has. at )\:= 0. 2n linearly independent 2w-periodic solutions. so called linear normal modes. For 0 < IKI << 1 and Izl << 1. we expect that the nonlinear system (3.1.4) is close to the linear system (3.1.7) and hence may have small amplitude periodic solutions with period near 21r near the periodic solutions of the linearized system (3.1.7). Furthermore. the equation (3.1.4) 2 = JAz + JB(A)z + JVN(z) has the linear part (3.1.9) 2 = [JA + JB(A)]z where the matrix C(A) = J(A + B(A)) is a smooth function of A. Since C(O) = JA has eigenvalues +i with multiplicity 28 and (—i) with multiplicty 2(n-2). C(A) will have an eigenvalue of the form C(A) i iw(A) for small IAI. where 0(0) = O, w(0) = 1 and 0. w are smooth functions of A. It may be possible to choose a particular parameter. say A1 with setting all the other X’s to zero so that by varying A1. a pair of eigenvalues of C(A) may vary either along the imaginary axis or across the imaginary axis. In two degrees of freedom case it turns out that the above choice is possible to examine the behavior of the periodic "‘3_L.. fl..- _.. 47 orbits of the nonlinear system (3.1.4) as A1 varies across zero. §2. Liapunov-Schmidt Reduction Now. we want to study the behavior of periodic solutions of (3.1.4) as A varies by the method of Liapunov-Schmidt Reduction in the presence of symmetry. Consider the system (3.1.4) again! (3.2.1) 2 = JVHA(z) = JAz + JB(A)z + JVN(z). We introduce the time scale. Set (3.2.2) t = pr. for Iu-II << 1. Then. in the new time scale T, (3.2.1) becomes (3.2.3) gé- u [JAz + JB(A)z + JVN(z)] JAz + (u-1)JAz + uJB(A)Z + quN(Z)- Hence, a 217-periodic solution of (3.2.3) corresponds in one to one manner to a 2wu - periodic solution of the original equation (3.2.1). So. henceforth. we look for 2w-periodic solutions of (3.2.3). Set (3.2.4) 2 = eJATu 48 in (3.2.3). where u 6 Rzn. Then, in the new coordinate u. (3.2.3) becomes (3.2.5) g; = (“_1)JAu + Me_JaTJB(7\)eJaTu + u e—JATJVN(eJATLI). Now. we claim that JVH§(eJaTu) = eJaTJVHé(u) and JVN(eJATu) = eJATJVN(u). More generally. we show the following Lemma: Lemma 3.2.1: Suppose H(z) = H2(z) + N(z) . z = (x,y) 6 R2n is in n 1 x? + 6 - - E x? + % ( J yJ) 2 j=e+1( J yJ) normal form with respect to H2(z) = l- N) J “Mo: 1 l-zTAz. i.e., H 6 ker ad . Then, the Hamiltonian vector field XH(z) 2 H2 JVH(z) is equivariant under the action of the one-parameter group of symplectic diffeomorphisms generated by the flow of XH (2). that is. 2 (3.2.6) x (eJAtz) = eJAtx (2). H H . . . . 1 T Proof Since H(z) is in normal form With respect to H2(z) = §-z Az. H 6 ker ade i.e., adeH = 0. Since XH2(z) = adH2(z). the flow generated by XH2(Z) is exp tadez = (exp tJA)z . Also. note that (exp tadH ) H(z) = H((exp tadH )z) (see . 2 2 Theorem 1.1.20). But. (exp tadH ) H(z) = H(z) since adH H = 0. Hence. 2 2 49 H((exp JAt)z) = H(z). That is, H(z) is invariant under the S1 action of the one-parameter group {exp JAtI t 6 S1}of symplectic diffeomorphisms. Now. XH((exp JAt)u) = J vzH(z)| z = (exp JAt)° u J [6H(exp JAtu) QEJT 6u 62 T = J ° {Qgfigl-° exp (-JAt)] (since H(exp JAt u) = H(u)) z) . [exp JAt - qu(u)] (since(exp (—JAc))T = eXp JAt ) exp JAt [J ° VuH(u)] ( exp (- JAt)J exp JAt = J since exp JAt is symplectic) exp JAt XH(u). Therefore. XH(exp JAt ° 2) = exp JAt XH(z). /// Returning to equation (3.2.5). by the Lemma 3.2.1. equation (3.2.5) can be written as (3.2.7) gg-z (p-l)JAu + pJB(A)u + quN(u) (u-l) JvH2(u) + u JvH§(u) + u JvN J V[(u-1)H2(u) + uH31u) + uN(U)]- 50 Note that the right hand side of the above equation is still equivariant under exp JAT. Now. we look for 2w-periodic solutions u(T) of (3.2.7) via the Liapunov-Schmidt Reduction with symmetry. Lemma73.2.2.2 The bifurcation function for (3.2.7) is just the right hand side of (3.2.7), i.e.. (3.2.8) V(a.u.A) = (p—l)JAa + pJB(A)a + quN(a). a 6 m2“ and hence V(a,u.A) inherits the symmetry from (3.2.7), i.e.. (3.2.9) V(eJATa.u.A) = eJAT V(a.u.A). proof Consider the linearized equation of (3.2.7) at u = 0. u = 1. A = 0. G = o. This equation has the 2n-periodic solutions u = constant. Let ¢(t). and W(t) be the 2n x 2n matrix whose columns are linearly independent 2n-periodic solutions of u = 0 and its adjoint equation respectively (the same as u = 0 in this case). Then. ¢(t) = W(t) = I2n= 2n x 2n identity matrix. For any f 6 G;W(R. R2n). define the projections P. Q onto the space of 2w-periodic solutions of u = 0 and its adjoint equation by 1 2w * - 2w * Pf = Q1 = ¢(t) b = I2n( [ ¢ ¢) (f ¢ f) o o 51 1 2n = §;-f f(t) dt. (see Hale [28]) 0 Then (3.2.7) is equivalent to (1'?) [(u-1)JAu + uJB(A)u + quN(U)] { (I-P)u Pfi P[(u—1)JAu + uJB(A)u + quN(u)] or. equivalently, (a) u (b) 0 a + K(I-P) [(u-llJAu + HJB(A)U + uJVN(U)] (3.2.10) { P[(H"1)JAu + uJB(A)U + quN(u)]. where K: (I-P) Cgv 7'(I-P)ng such that Kg is the unique 2w-periodic solution of (1 = g(t) for g 6 (I-P)C;Tr with PKg = 0 and a 6 R2n such that a = Pu. Let F(u,a.u.A) = u - a - K(I-P)[(u-1)JAu + uJB(A)u + uJVN(u)]. Then. F(0.0.1.0) = 0 Fu(0.0.1.0) = I2n' Hence. by the Implicit function theorem. there exists a unique function u* = u*(a.u.A) for Ial << 1, Iu—ll << 1. IA] << 1 such that F(u*(a.u,A). a.u.A) = 0 and u*(0.1.0) : 0. But. notice that u* = a satisfies (3.2.10)(a). By the uniqueness of the Implicit function theorem. it follows that (3.2.10)(a) has unique solution u* = a. Substituting u = a into (3.2.10)(b), we obtain the bifurcation equation. 0 = P[(u-1)JAa + uJB(A)a + HJVN(a)] = (u-l)JAa + uJB(A)a + pJVN(a). 52 Therefore, the bifurcation function of (3.2.7) is V(a.u.A) = (Lt-1) JA a + uJB(A)a + quN(a) JV[(u-1)H2(a) + urge) + u N(a)]. a 6 IR Right hand side of equation (3.2.7). 2n That is. the solution set (a.u.A) of V(a.p.A) = 0 is just the critical points of equation (3.2.7). Moreover, since the Right hand side of JAT (3.2.7) is equivariant under e , clearly V(a.u,A) is also equivariant under exp JAT. /// Remark: Since V(a.u.A) can be expressed as (3.2.11) V(a.p,A) = JVS(a.p.A) where S(a,u,A) = (u—1)H2(a) + qu(a) + u N(a). it follows that finding zeros (a.u.A) of V(a.u.A) = 0 is equivalent to finding critical points of the real-valued function S(a.u.A) and each zero (a.u.A) of V(a.u.A) = 0 corresponds locally in 1 — 1 fashion to each 2w—periodic solution z(T) = (exp JAT) ° :1 of (3.2.3) and so locally 1 - 1 corresponds to each 2wu - periodic solution of the original equation (3 2.1). From now on. we try to find the zero set of V(a,p,A), i.e.. the critical points of the real-valued function S(a.u.A). Note that V(O,p,A) = 0 for all p z 1. A z 0 2 DaV(0.1,0) = (u—1)JA + pJB(A) + “JD N(a) a=0. u:1,A=0 = 0 where N(a) = 0(Ial4) 53 implies (a.u.A) = (0.1.0) is a singularity of V(a.u.A) and so is a possible bifurcation point. Furthermore. D V(O.1.0) = pJD3N(a)| = 0, aa a=0 DaaaV(O.1,0) = pJD4N(a) l 2 o. a=0 So. V(a.u.A) = 0(Ial3) as a 6'0, at u = 1 and A = 0. Now, returning to bifurcation equation (3.2.8). V(a.u.A) = (u-1)JAa + uJB(A)a + quN(a). x we first try to determine u = u (a.A) uniquely and continously so that * def ~ V(a.u (a.A).A) = V(a.A) is orthogonal to JAat We put F(a,u.A) = 1 < JAa, V(a.u,A) > for a ¢ 0 (3 2 12) IJAaIZ ' ' _ ”‘1 + u < JAa, JB(A)a + JVN(a) >. lJAal2 Then we have F(O.1.0) = 0 and Fu(0.1.0) = 1. Hence. by the Implicit function theorem. there exists a unique C1 function 11 = u*(a.A) near a = 0. A = 0 such that F(a. u*(a,A) .A) = o, “(0.0) = 1. i.e.. 54 (3.2.13) < JAa, V(a,u*(a.A). A) > = o for o < |a| << 1, [AI << 1. In fact. from (3.2.12). < JAa, JB(A)a + JVN(a) > lJAal2 u(1 + ) = 1- * Hence. u (a.A) is explicitly given by IJAalz IJAaI2 + < JAa. JB(A)a + JVN(a) > lal2 |a|2 + < Aa. B(A)a + VN(a) > 1 1 + 0(IAI + lalz) So, for IaI << 1 and IAI << 1. u*(a.A) z 1 - 0(IAI + laIZ). u*(a.A) (3.2.14) Notice that even though the formula (3.2.14) may be valid for all a,A, we must restrict ourselves to a sufficiently small neighborhood of (a.A) = (0,0) to ensure that Iu*(a.A) - 1| << 1. Lemma 3.2.3: Let v(a,A) = V(a.u*(a.i), A) for all 0 < lal << 1 and IA] << 1. where u¥(a.A) is give by (3.2.14). Then, u*(a,A) is invariant and V(a.A) is equivariant under the action of the l—parameter group {exp JAt: t e R}. that is. u*(exp JAt a.A) = u*(a.A) V(exp JAt a.A) = exp JAt W(a,A). proof Let h(a,A) = JB(A)a + JVN(a) in (3.2.14). Then from (3.2.14). 55 u*((exp JAt) a.A) = I(exP JA‘) alz . Iexp JAt a|2 + < JA (exp JAt) a. h(exp JAt a.A) > I(exp JAt) aI2 = < (exp JAt) a. exp (JAt) a > = < a. (exp -JAt) ° (exp JAt) a > = Ialz. Since h(a.A) is equivariant under eJAt, < JA exp JAt a. h((exp JAt) a.A)) < (exp JAt) JAa. (exp JAt) h(a.A) > < JAa. h(a.A) >. x * Hence. u (exp JAt a.A) = u (a.A). Also. 3(exp JAt a.A) V((exp JAt) a. u*((exp JAt) a.A), A) V((exp JAt) a. u*(a.A), A) (since u* is invariant) (exp JAt) V(a. u*(a.A), A) (since V is equivariant) (exp JAt) W(a.A). /// Note that for o < |a| << 1, lu-ll << 1, if V(a.u.A) = 0 then F(a.u.A) = 0 and by the uniqueness of u* we must have u = “*(a.A) with u*(0.0) = 1. So. V(a.u.A) = 0 iff V(a.A) = V(a.p*(a.i). A) = 0. Hence. (a.A) is a zero of V(a.A) = 0 iff (a.u*(a.A). A) is a zero of V(a.u.A) = 0 iff (a. u*(a,A), A) is a critical point of equation (3.2.7) iff z(T) = exp JAT a is a 2w—periodic solution of (3.2.3) iff z(t) (exp JAt/u*(a.A)) a is a 2w p*(a.A) — periodic solution of the orginal equation (3.2.1). 56 §3. Reduction to a_gradient system Now. our problem to study the periodic solution of (3.2.1) near those of the linearized equation is reduced to finding the zeros of the bifurcation equation V(a.A) 0 which is the Zn x 2n finite system. Furthermore, since V(a.u.A) JVS(a.u,A) by (3.2.11) Where S(a.u.A) = (u—1)H2(a) + uHé(a) + uN(a) = qu(a) ‘ H2(a)- we can easily express V(a.A) as a gradient-like system as above. Lemnla 3.3.12 Let S(a.A) = H2(a) - 11*(a.A) [Hk(a) - c]. for any constant G. Then JV(a.A) = VS(a.A) on the energy surface HA(a) = c. proof V(a.u.A) = Jv[uH3(a) - H2(a)]. JV(a.u.A) v1H21a) - when VH2(a) - u V(HA(a) - c] for any fixed constant c. If u = u*(a.A). then v1H.(a) — flex) . (We) - c)1 vH21a) - flex) «11%) — c) — vu*(a.A) - (We) — c) VS(a.A) JV(a.u*(a.A). A) on HA(a) = c JV(a.A). /// Remark: This Lemma 3.3.1 (and Lemma 3.3.2 in the following) are due to Chow and Mallet-Paret [14]. but in our case these Lemmas are trivial consequences of the fact that our Hamiltonian function is in normal m with respect to H2 and so the bifurcation function V(a.u,A) is just the right hand side of equation (3.2.7). which is again. a 57 Hamiltonian vector field. Now. by Lemma 3.3.1, the problem of finding zeros of V(a.A) = 0 is again reduced to find the critical points of the potential function S(a.A) = H2(a) - u*(a,A) ° [HA(a) - c] on the energy surface Hk(a) = c. Thus. we must solve the two equations VS(a.A) = VH2(a) - u*(a.A)VHA(a) = 0 (3.3.1) >\ H (a) = C simultaneously for each given A and c. Let 3(A,c) be a critical point of S(a.A) on the energy surface Hk(a)=c. Define 6: m2“ x m x m e m2“ x m by [612121) — u*(a.A)VH>‘(a)] G(a,A,c) = A . H (a) - 0 Then. we know that G(0.0.0) = [g] = O. DaG(O’O'O) = 0. Hence (A,c) = (0.0) is a possible bifurcation value for the zeros of the system (3.3.1). Note that on each energy surface Hk(a) = c. VS(a.A) = VH2(a)- u*(a.A)vHA(a) and this resembles the Lagrange multiplier method when we compute the critical points of H2(a) with the constraint Hk(a) = c in which case we solve the equation VH2(a) - n° VHA(a) = 0. e.g., for a in term of (n.A) and then using the constraint Hk(a) = c we determine n = n(A.c) and so determine a = a(A,c). In the following Lemma. it turns x out that the Lagrange multiplier n so obtained coincides with u (a.A). Lemma. 3.3.22 For each IAI << 1 and [CI (< 1. let a = 5(A.c) be a nonzero critical point of the real-valued function g(a.A.n) = H2(a) - n°HA(a) with Hx(a) = c. Then. n = u*(5.A) in a sufficiently small neighborhood of (0.0). proof Since a = 5(A,c) is a critical point of g with Hk(a) = c for each A. c. we have _ A _ VH2(a) - nVH (a) = 0 where n = n(A,c). Hence, vS(£.A) = vH2(a) - u*(5,A)vHA(5) = [n - u*(§.A)]° VHA(5). Then. from (3.2.13) and Lemma 3.3.1, we have < JAE. V(a.A) > 0 II = < 1A5. J’lvs(5.i) > on 3*(3) = c * — - A - . —1 = - [n - u (a.A)] < JAa. JVH (a) > (Since J = - J). Multiplying both sides by u*(§1A). we can write at- —x»)\— _ _ 0 = -[n - u (a.A)] ° < JAa, u JVH (a) — JAa + JAa > Since V(a.A) = ”*JVHA(a) - JAa and < JAa. V(a.A) > = o for all 0 < |a| << 1 and IA] << 1, we have 59 0 = - [n - u*(5.>\)] . < JAE. V(a.A) + JAE > * ... _ = - [n - u (a.A)]° Ial2 Hence. n = “*(EZA) since 5'; 0. /// Therefore, finally our problem to study the periodic solutions of (3.2.1) has been reduced to finding the critical points of the real -valued function g(a.A.n) = H2(a) - n ° HA(a) with Hx(a) = 0. So. if we solve the equation V(H2(a) - nHA(a)) = 0 with Hx(a) = c then n will be automatically determined as ”*(a.A). Note that g is invariant and Vg is equivariant under the action of the group {eJAtz t 6 R) and hence if 5': 5(A,c) is a solution of Vg = 0 with HA(a) = c. then eJAtE are also critical points of g on the same energy surface for all time t. Now. we summarize all the above results in the following theorem. which will be a main theorem of this thesis. Theorem 3.3.3: Consider a family of Hamiltonian functions HA(2) = H212) + H§(z) + N(z). z = (x.y) e m2“ passing through 12 ... 112 -1: ... I -1. semisimple resonance at A = 0. where H2(z) is given by H2(z) = %- g (x2 + yz) — §- 2 (x2 + yz) - é-ztAz jzl J J j=3+l J J 60 and H§(z) = é-ztB(A)z is a versal deformation of H2(z) and N(z) is a higher order term. Suppose that HA(2) is in Hz-normal form. Then, the periodic solution of the Hamiltonian system i = JVHA(z) - c are locally in a one-to-one on the energy surface HA(z) correspondence to the critical points of the real—valued function g(a.A.n) = H2(a) - n . H*(a) c. More precisely, if a": g(A.c) is a critical point of g c for Ial. IAI. Icl << 1. then the Hamiltonian system has a on Hk(a) on Hk(a) periodic solution z(t) = eJat/u*(2_1-.A). 2-1- with period 2nu*. where u*(a.A) is given by 2 u*(a.?\) = lal |e|2 + < Aa, B(A)a + VN(a) > for Ial, IAI << 1. Obvious from all the Lemmas in this chapter. /// proof: Hence. from now on. we concentrate only on the problem to find critical points of the real-valued function 61 ‘ A g(a.A.n) = H2(a) - n H (a) A on the energy surface H (a) = c. By using the invariance of g and the equivarience of Vg under the action of (exp JAt It 6 R}. the problem to solve the equation Vg - 0 can be further reduced and we are going to work out this problem in the two degrees of freedom case explicity to show the bifurcation of the periodic orbits. 62 CHAPTER 4: TWO DEGREES OF FREEDOM 1: - 1 SEMISIMPLE RESONANCE PROBLEM In this chapter. we apply the general theory of Chapter 2 and Chapter 3 to the Hamiltonian function of 2 degrees of freedom with the nonpositive definite quadratic form at 1: -1 semi-simple resonance. and study' the 'bifurcations of jperiodic orbits as the 'parameter passes through the resonance. §1. Normal form and Versal deformation Consider the Hamiltonian H: R4 -> R. Coo, with the nonpositive definite quadratic form at 1: -1 semisimple resonance: 1 (X? + y?) - g(xg + Y2)- twee (4.1.1) H2(X,Y) : First, we find the normal form of H with respect to H2 up to the 4th order. ‘In this case. the map ad : Cm(R4. R) e>Cm(R‘. R) is given by H2 2 6H 6 6H 6 ad = 2 ( 2 - 2 ) H . 6.6. 6.6. 2 J=1 yJ XJ x1 ya 6 6 6 6 = (Y15;:'- X155?) — (Y25;;'- X2 )- aY2 63 In complex conjugate coordinates (z.§) = (Z1, 22, 21. 2;) 6 $4 with zj= Xj + iyj (j = 1.2). Hz and adH can be rewritten as 2 ~ H2(x.y) = H2(z.2) = Z1 23 - "Zz z2 knee 811%’ (2,2) = -i (216—:- — 5-1—2“) -’ (22% "' .2— i 2 1 21 2 622 ) The action of adfi on the basis monomial zkée for the space @n(z.2) of 2 homogeous polynomial of degree n in 2.2 is ad§2(zkée) = —i[(k. — e.) — (k2 — 22)] (zkie). Hence. zkEe 6 ker adfi iff k1 - 81 = k2 — 82 (resonance relation). 2 It follows immediately that there are no third or hdgher odd-ordered terms in the normal form of H. The (Hilbert) generators for Ker adH are given by 2 2 2 (4-1-2) p1 = Iztl2 X? + yf. 02 = lzel2 = X3 + YE. X1X2 ‘ Y1Y2 . 94 = Im(z122) = X1Y2 + X2Y1. P3 = Re(zizZ) Also. the generators for Ker ad are given by 64 62 = Re (2% Z2 21) = (XI + YI)(X1X2 ‘ Y1Y2) es = Im (2% z2 21) = (X? + YT)(X1Y2 + X2Y1) 64 = zizzgigz = (X? + YI)(X§ + yg) 95 = Re(zfzg) = (XI “ YI)(X§ ' Y3) ‘ 4X1X2Y1Y2 (4-1-3) 66 = Im(ZfZ§) = 2X1yt(X% - y%) + 2X2y2(Xf - y?) e7 = Re(ziz§22) = (X2 + Y§)(X1X2 ‘ Y1YZ) Im(zizf22) = (X2 + yg)(X1YZ + X2Y1) (D a; II e9 = 23—3 = (X5 + yg)2, Therefore, the normal form for H(x.y) with respect to H2 up to the fourth order is given by (4.1.4) H(x.y) = H2(x,y) + H4(x,y) + (higher order terms) 9 where H4(x.y) = 2 ajej and ej’s are given in (4.1.3). Moreover, from 1:1 the general theory of Chapter 2, the versal deformation of H(x.y) up to the second order i.e.. in the space 92(x,y) can be written as A A 5 (4-1-5) H (X-Y) = H2(X:Y) + H2(X.Y) + H4(X.Y) + 0(IX-YI ) where A 1 2 2 1 2 2 H2(X-Y) = E >\1(X1 + Y1) ‘ 5 >\2(X2 + Y2) + 7\:3(X1X2 ‘ Y1Y2) + 64(X1Y2 + X2Y1)- Now, we consider the truncated Hamiltonain. denoted again by Hk(x,y), containing only a single fourth order term e3. 65 (416 mRJ)=%UJ)+@(Kfl+HJLW with fi4(X-Y) = e3 = (X? + Y?) (X1Y2 + X2Y1)- Remark: Here. we picked up a fourth order term e3 randomly just for simplicity of calculation to show our method to get the bifurcation explicity in the presence of nonlinear terms. Even if we consider the full nine fourth order terms in H.(x.y). our methodology will be just the same except a slightly more involved computation. 4 Rewriting (4.1.6) in vector—matrix notation with z = (x,y) 6 R . we have (4.1.7) H*(z) = H2(z) + H§(z) + fi.(z) = é-ZTAZ + %- TB(A)z + fi.(z). where A = diag (1. -1, 1, -1), and P A1 A3 | o A.) A3 -A2 | A4 0 O 64 l >\1 "ks . A. o |-A3 -A2 B(A) = The corresponding Hamilton's equation is (4.1.8) 2 = JVHA(z) = JAz + JB(A)z + Jvfi.(z). where 66 | 1 01 o A. | A1 -A3‘ 0 o -1 A. o I—A3 -A2 JA = -1 OT 0 ’ JB(A) = '61 ‘63 I O ‘64 0 1| 'Aa >\2 I-Aq O L P 2Y1(X1Y2 + X2Y1) + X2(XT + Yf)l _ X1(XT + Y?) JVH4(Z) = -2x1(x1y2 + X2Y1) _YZ(XT + Y?) ‘Y1(XI + YT) Remark: Since each term in the truncated Hamiltonian (4.1.7) is in the normal form with respect to H2. HA(z) is invariant under the action (rotation) of the one-parameter group of symplectic diffeomorphisms {exp JAt: t 6 R) generated by the flow of XHZ. in Chapter 3, XHA(z) = JvHx(z) is equivariant under the same action. Hence. by Lemma 3.2.1 Furthermore. since {H2.HA) = O, the system (4.1.8) is completely integrable with integrals Hx(z) and H2(z). Also note that the system (4.1.8) is a versal deformation of H in 92(R‘.R) with codimension 4 of the unperturbed system 2 = JVHO(z) preserving the Hamiltonian character. Since the number of parameters are too many to examine the qualitative behavior of (4.1.8). we are going to restrict ourselves to the codimension one bifurcations by choosing a suitable parameter and setting the other parameters to be zero. §2. Invariant manifolds of the linearized system The linearized Hamilton’s equation of (4.1.8) at z = O and A = O 67 (4.2.1) 2 = JAz with the solution z(t.zo) = (exp JAt) - zo starting from the initial point 206 R4 at t = 0. Since the 4 x 4 matrix JA has the eigenvalues t i each with multiplicty 2 and exp JAt = I(cos t) + JA(sin t), we have Iz(t)l = Iexp JAt - zol = [20]. Hence each solution curve z(t,zo) = (exp JAt) ° 20 is a 2w—periodic circle lying on the 3-sphere S31 x? + 2 . 4 y? + x3 + y3 = Izol in R . Notice that the linear system (4.2.1) has the Hamiltonian 1 1 (4.2.2) H(x.y) = §(xf + y?) — §(x§ + yg) which is the sum of the energy functions of two harmonic oscillators running opposite in time both. with frequency 1. Furthermore. the system (4.2.1) has another integral 1 1 (4.2.3) L(x y) = g(xf + y?) + 5 (x3 + yé) as we can easily see from the fact that (H,L) = O. The function L(x.y) may be considered. up to canonical change of coordinates, as the angular momentum of these oscillators. Now. we may consider the so-called "energy — momentum mapping" H x L de (H,L) : m4 e m2 68 defined by (H X L)(Xi.x2.yi.y2) = (H(XJ): L(X-YD- Then, each orbit. i.e.. 2w-periodic circle. of the linear system (4.2.1) lies on the level set of the mapping H x L. (4.2.4) (H x L)'1(h.e) = {(x,y) e m4 H(x.y) = h. L(x.y) = e) where h 6 R. 8 2 0. If (h.8) 6 R2 is a regular value of the mapping H x L. then the level set (H x L)-1(h,€) defines a smooth 2 - dimensional invariant manifold of the system (4.2.1) in R4. However. if (h.8) 6 R2 is a critical value of the mapping H x L. that is. for some 20 6 (H x L)—1(h.£). the derivative D(H x L)(zo): Tz R1» T(h e)R2 is not surjective. then the level set (H x IJ_1(h.e) o 9 will be at most a 1 — dimensional critical manifold. To be more precise, let’s find out the critical sets of the mapping H x L. Recall that a point z = (x1,x2.y1.y2) 6 R4 is a critical point of H x L iff D(H x L)(z) is not surjective. Since D(H x L)(z) = (DH(z). DL(z)). z 6 m4 is a critical point of H x L if (i) DH(z) = O or DL(z) = O or (ii) DL(z) + ADH(z) = 0 for some A ¢ 0. i.e., z is a critical and A is a Lagrange multiplier. (11) point of L H—l 69 In the case (i), we have only the trivial critical point z = O with the energy H = 0 and the momentum L = 0. In the case (ii). for each h 6 R. the solutions of the system of equations for A ¢ 0, (1 + A)X1 = 0 (I - K)X2 = O (1 + A)Y1 = O (1 ‘ A)Yz = 0 yield the critical circle 31 = {(x1.0,y1.0) 6 m4] g(xf + y?) = h} for h > o and 4 1 2 2 s2 = {(O,x2.0,y2) e m I §(x2 + y2) = —h} for h < o with the corresponding critical values (H,L) = (h,h) for h > O and (h.-h) for h < 0 respectively. For h = 0, (ii) yields only the trivial critical point z = 0. Therefore. for those critical values (h.8) with 8 = h for h > 0 and 2 = -h for h < O of the mapping H x L. the level set (H x L)_1(h,8) is the 1 — dimensional circle lying in the (x1,y1) plane for h > 0 and in the (x2.y2) — plane for h < 0 respectively. The foliation of the constant energy surface H_1(h) for each given h 6 R with respect to the parameter values of 8 can be easily examined by using polar coordinates. Putting x1 = 71 cos 61. y1 = 71 sin 91 x2 = 72 cos 62 y2 = 72 sin 92, then the level set -1 1 H (h) = {(X1.xa.y1.y2) 6 R g(Xf + y?) - §{X§ + YE) = h} 70 can be expressed as > o. h e m). H-1(h)‘= {(71:72) 6 mzl é’”? ‘ é’Tg = h- 71.72 which is a hyperbola for each h ¢ 0 and a straight line for h = 0 in the (11.72)-p1ane as shown in < Figure 4.2.1 >. T2 . < Figure 4.2.1 > For h ¢ 0. a constant energy surface H_1(h) is diffeomorphic to S1 x R2 and hence is not compact. while for h = 0. it is a cone (0) x R2 over S1 with vertex at the orgin. Also. the level set " 1 ~ 4 1 2 2 1 2 2 L (3) = {(Xi.x2.yi.y2) € R §(x1 + yi) + §(X2 + Y2) = 3} can be rewritten as 71 -1 2 1 2 1 2 L (E) = {(71.72) E [R E71 + 572 = e, 71.72 2 0. e 2 O}, which is a quater-circle in the (w1.72)-plane with radians v5? as shown as dotted lines in < Figures 4.2.2 >. Hence, the level set (H x L)“1 (h,£) may be expressed as the set of intersection points of the two curves 4 I 4 M II 2 2 2h (h e m) 7% + 13 22 (e 2 0) in the (71.72) —plane as is shown in < Figure 4.2.2 > < Figure 4.2.2 > Hence, for h > o, if e > h then (H x L)‘1(h.e) = 31 x s1 = T2 and if e = h (critical values of H x L) then (H x L)-1(h.£) = S1 x {O}, which is a circle lying in the (x1,y1)-p1ane. While for h < O. (H x L)—1(h.e) = s1 x s1 = T2 for e > -h and (H x L)‘1(h,e) = {0} x 31 for e = —h (critical values of H x L). If 0 g e ( Ihl then (H x L)_1(h,€) = ¢ for 72 |h| g o. 2 T for e > O For h = o, (H x L)"1(h,e) = { {O} for e = 0 Therefore, we can conclude that every solution curve z(t) = (exp JAt) 20 of the linear system (4.2.1) is a (Zn—periodic) circle lying on S1 x {O}, {O} x S‘, or T2 depending on the values of h and 9. Remark: As van der Meer did in [35], we may also use the Si—invariant variables defined by W1 = (X? + y?) - (XS + y?) = 2h (fixed) W2 = (X? + y?) + (X? + yi) = 29 2 0. ”a = 2(X1X2 + Y1YZ) "4 = 2(X1Y2 ‘ X2Y1) with relationship W? + fig + WE = w: in order to describe the foliation. of the constant energy surface H-1(h) for given h 6 R. Rewriting the identity as NS — wg — WE = 7? = (2h)2 = constant, wz = 28 2 0, then the mapping 73 F: (X1:x2:Y1:YZ) " (772,778’7”) With 772 Z O 1 maps the constant energy surface H_1(h) = S x R2 to a connected piece of two-sheeted hyperboloid in R3 #3 - «g — vi = (2h)2, 72 2 0 if h ¢ 0 and to a half—cone if h = O. The intersection of the hyperbold and the plane #2 = 22 2 O is a circle whose preimage under the mapping F is F_1(28,w3,w4) = ( ll x: L)-1(h,8) = toroidal energy' momentum surface. §3. Eigenvalues of the perturbed linear system From the above global analysis in Section 2 about the linear flow (4.2.1), we may expect that for [2' << 1 and [AI << 1. the family of nonlinear Hamiltonian systems (4.3.1) é = JVHA(z) = JAz + JB(A)z + Jvfi4(z) is close to the linear system (4.2.1) and hence may have small amplitude periodic solutions with period near 21r near the periodic solutions of the linear system (4.2.1). Now, the linearized equation 74 of (4.3.1) at z = O is (4.3.2) 2 = J(A + B(A))z dgf JA(A)z. where ’ 0 A4 | 1+>\1 —A3 ‘ A. o l -k3 —1—A2 JA(A) = -1-x1 -x3| o —x. L_A3 1+A2| “A4 0 l After a tedious calculation of the characteristic polynomial of JA(A), we find that (4.3.3) det (aI - JA(A)) = a4 + a2[(1 + x1)2 + (1 + h2)2 — 2x3 — 2kg] + [R3 + Ki + (1 + K1)(1 + k2)]2, and the eigenvalues are given by (4.3.4) (12 = - £0 + 2(1)? + (1 + A2)? - 2A3 — ZAE] i 52 + x1 + Ann/(A.- m2 - 40% + N37. From (4.3.4), we notice that (i) when A1 = A2 = A3 = A4 = O, the eigenvalues of JA(A) are a = i(double), a = -i(double) as they should be. (ii) when k3 = K4 = 0, (4.3.3) becomes a‘ + a2[(1 + (1)2 + (1 + )2)2] + (1 + A.)2(1 + x2)2 = [a2 + (1 + >\1)2] [0t2 + (1 + 7\.2)2]- 75 So, the eigenvalues of JA(A) are a = i i(l + A1), a = i i(l + AZ). (iii) when A1 = R2 = 0, (4.3.4) becomes (:2 = - 52 — 27x3 — 24%] i 21:53 + A? = (—1 + x3 + ii) i 21:53 +‘ii‘ = (1X3 + Xfi—i i)2. So, the eigenvalues of JA(A) are a = i (e + i), a = i(e - i) where e - Jig—Jar. Since we are mainly interested in the cases when the eigenvalues vary along the imaginary axis, or across it, from now on, we restrict ourselves to the following two cases: Ca.se(a) >\2=)\3=7\4=0 (01‘7\1=)\3=7\4=0). In this case. the eigenvalues of JA(A) are i i. ii(1 + A1). i.e. the double eigenvalues i i of JA(O) = JA split along the imaginary axis. Case (b) A1 = A2 = A3 = O (or A1 = A2 = A4: 0). In this case, the eigenvalues of JA(A) are i(7\4 + i), i (h4-i), i.e. the double eigenvalues i i of JA(O) = JA split aggggg the imaginary axis. 76 < Figure 4.3.1 > Note that in each case (a), (b), our Hamiltonian Hk(z) takes the form 7x 1 1 1 H (Z) = g(X‘? + Y?) - g(X‘Z + y?) + @100? + y?) + (X? + yf). (X1Y2 + X2Y1) K 1 1 H (z) = §(X¥ + Y?) ‘ §(x§ + YE) + A4(X1Y2 + X2Y1) + (X? + Y7)(X1Y2 + X2Y1) respectively. Egggggi By using the same method as in Section 2. it may be possible to study the bifurcations of invariant manifold (HA x H2)_1(c,m) as A1 (or A4) varies for various values of c and m. But, this does not give any detailed informations about the bifurcation of periodic orbits lying in those invariant manifolds. Hence. we will have to examine the 77 local bifurcations of the periodic orbits themselves by other means. In the following, we will do this by using the method described already in Chapter 3. In van der Meer's thesis [35], he examined the bifurcations of invariant manifold of the Hamiltonian system with the quadratic part H2 = (x1y2 + xzyi) + é—(xf + X?) which is 1: —1 nonsemisimple case. He considered the energy—momentum mapping H x S where S is the semisimple part of Hz and obtained the standard form C x S and its unfolding GD x S by using singularity theory and finally examined the fibration (GD x S)_1(g,s) as v varies for given g,s. §4u Local bifurcations of periodic orbits as the eigenvalues split along the imaginary axis Now, we follow the methods described in Chapter 3 with Hamiltonian (4.4.1) Hx(z) = zTAz + 2TB(A)Z + H;(z), [\DIH [\DIH where A = diag (1, «1, 1, -1), B(A) = diag (A1, 0, A1, 0), and H4(z) = (X? + YT)(X1YZ + X2Y1)- The corresponding Hamiltonian system is (4.4.2) 5 = JVHA(z) = JAz + JB(i)z + Jvfi;(z), 78 where the linear part A(A) = A + B(A) has eigenvalues, i i, and (1 +)\1)i. After introducing the time scale t = 111', I11 - 1| << 1, (4.4.2) becomes (4.4.3) gé-z u[JAz + JB(K)z + JVH;(Z)] . . . . . AT 4 and after putting this into rotation coordinates z = e‘J u, u 6 [R , equation (4.4.3) becomes (4.4.4) %%'= (p — 1)JAu + uJB(h)u + uJVH;(z) by Lemma 3.2.1. And the bifurcation function of (4.4.4) is given by (4.4.5) V(a,u.i) = (u - 1) JAa + uJB(A)a + qufi;(a). a e m4 with the equivariant property eJAT V( a, p,i) = eJATV(a,u,h) by Lemma 3.2.3. Also, we can choose p = u*(a,h) uniquely and continuously so that < JAa, V(a,u*(a,i),x)> = o for all 0 < |a| << 1, lil << 1. In fact, from (3.2.14), u*(a,h) is given by 79 (4.4.6) u*(a.h) = lalz ._ |a|2 + < Aa, B(A)a + vH,(i) > With B(A) = diag(k1, 0, A1, 0), (4.4.6) becomes 2 (4.4.7) u*(a,k) = lal Ialz + h1(af + 3%) + 2(af + a§)(aia4 + a233) 1 1 = = = 1 - 0(I7~1I+IaI2) 1 + A1 P(a) + 2P(a)Q(a) 1 + 0(IA1|+|3|2) asbJeOaMIMIAQ a2 + a where p(a) = 1| I 3 0 g p(a) g 1 for any a ¢ 0, a 2 q(a) = (a1a4 + azaa) = O(|a|2)° Further, letting V (a.A) = V(a,u*(a,i),i), then by Lemma 3.2.3, V(a.A) JAt and each zero (a.A) of V(a.A) is locally is also equivariant under e . * . . . in a one to one correspondence to the 21711 (a,?\) - periodic solution z(t) = eJAt/u*(a,>\) each zero (a.A) of V(a.A) is a critical point of the scalar-valued a of our system (4.4.2). Also, by Lemma 3.3.1, function (4.4.8) S(a,h) = H2(a) - u (a,h)°[H (a) - c] on the energy surface HA(a) = 0. Moreover, by Lemma 3.3.2 in Chapter 80 3, we know that the solution of VS(a,N) = O with the constraint HA(a) = c can be obtained by solving Vg(a,h,n) = O with Hk(a) = c, where (4.4.9) g(a.x.n) = H2(a) - n . H‘(a) _ *— for a = a(h,c) with n = u (a.A). Therefore, we concentrate on solving the equation Vg(a,?\,n) = O for a = 50.1)) with n = u*(a,}\) given in (4.4.7). Then, by Theorem 3.3.3 this solution a(>\,n(7\,c)) will be *— locally in 1 - l correspondence to the 2wu (a.A)-periodic solution *. z(t) = eJAt/ll (a.A);- of the orginal equation (4.4.2). Now, g(a.x.n) = H2(a) - nH*(a) H2(a) - n . [H.(a) + H§(a) + fi;(a)1 1T 1 T — =(1-n)‘§aAa-n°§aB(>\)a-U°H4(a) 1 1 u—n)o§fi+a§—s—an—no§mm£+an- H(aI + ag)'(aia4 + 82a3)- Hence. the system of equations Vg(a,h,n) = 0 becomes 81 ’11) (1-")31 ‘ 774131 “ "[231 (3134+ 3233) + 21413-11a + a§)] = O (2) -(1-n)a2 -n[ae(af + 33)] = O (4.4.10)<(3) (l-n)ae ‘ ”4133‘ "[233(aiaq+ a233) + a2(af + 33)] = O (4) '(1-n)a4 - n[a1(a'f + 213)] = 0 together with the energy condition HA(a) = c. (5) gm? +)a§) - 3,1421: + a3) + $7444 + as) + (a? + as) (a.a.+ azaa = c. 5 Note that for each A1 and c,(4.4.10) is a system of 5 equations in 5 unknowns a1,a2,a3,a4, n and so we can solve (4.4.10) (1), (2), (3), (4), for a = 5(A,n) in terms of N1,n and make use of (4.4.10) (5) HA(a) = c to determine 7) = n(7\,c) and hence determine a = g(k,n(>\,c)). Furthermore n(k,c) will turn out to be n(h,c) = u*(alh). Therefore, by using a = EIK,n) and n = u*(alh) we can examine the number of solutions as the parameter A1 varies. Also. note that system (4.4.10) is equivariant under the rotation exp JAt for all t and in particular invariant under the reflection a1<—> a3, a2<—> a4. Clearly, a = O is a trivial solution of (4.4.10) for all N1 with c = O.For c ¢ 0, a = O is no longer a solution of (4.4.10). Recall that n = “*(a,A) z 1 - 0(IA1I +IaI2) as Ial 4’0 and [Ail » 0. Hence, for sufficiently small [Nil and la], we always have n > O and O ('n < 1 for A, > O and q(a) > O and n > 1 for A1 < O and q(a) < 0. Since u*(0,0) = 1, n cannot be zero. If n = 1, then we have K1(af + a3) + 2(af + a§)°(a1a4 4 azas) = O and hence our system 2 = JVHA(z) reduces to the linear system i = JVH2(z). Therefore we may assume n ¢ 1. Now, for n ¢ 0 and n ¢ 1, we can write (4.4.10) (2), (4) as 82 a... = - fiadaf + a4) (4.4.11) n 2 2 a4: ‘ T:E'ai(ai + 83). Substituting (4.4.11) into (4.4.10) (1), (3), we have, by the reflection symmetry a1 ee>a3, a2 ee>a4, (i) a1[(1-n) (l-n - nkl) + 3n2(a? + a§)2] = 0 (4.4.12) (ii) ae[(1-n)(1-n-nkl) + 3n2(a§ + a§)2] = 0- Note that if we can solve the system (4.4.12) for a1, as in terms of A1, n, then by (4.4.11), a2, a4 are automatically determined and so we can determine the solution of the system (4.4.10). Hence, the 4 x 4 system (4.4.10) (1) - (4) has been reduced to solving the 2 x 2 system (4.4.12), which is entirely due to the equivariance of the orginal system (4.4.10). Also notice that if N1 = 0 then (4.4.12) and hence (4.4.10) has only the trivial solution a = O with c = 0. Now, we consider several cases: case (i): a1 = O and a3 = O. This clearly satisfies (4.4.12) and from (4.4.11) we have a2: a4: 0 O. and from (4.4.10) (5), we have c Hence, we get trivial solution a = O for all h, with energy c = 0. case (ii) a1 = O and a3 ¢ 0 83 From (4.4.12)(ii), we have (l-n)(1-n - nk1) + 3712 a3 = 0- or, a. _ _ (1—n)(1—n - n1.) 3 — . 3n2 From (4.4.11), we have a2 = lfin a3, a4 = 0. Hence, in this case, we have solutions of (4.4.10) of the form =84=O _ (1-n)(1-n -nk1). a2 = ‘ n a3. a1 (4.4.13) ( l-n 3n2 a3: case (iii): a3 = O and a1 ¢ 0 By the reflection symmetry a1+—> a0, a2€_) a4, we have the solutions of (4.4.10) of the form a3 = a2 = O (4'4'14) l a? = _ (l-n)(1-n - nkl). a4 = - T95-a2. 3n2 case (iv): a1 ¢ 0 and a3 # O. From (4.4.12)(i)(ii), we have (1-n)(1-n - nk1) + 3n2(a? + a3)2 = 0 or, mf+£V=_(PMUm-nhl 3n2 Therefore, it follows from the above cases that the most general solution of (4.4.10) including cases (i)(ii)(iii)(iv) can be written as 84 r(a¥ + 83):? = _ (I‘TIHI‘TI - URI) (a) 3n2 (4H415) < a2 = _ 1:7.” 83(83 + a3) (b) a4 = ' I?” a1(af + 33) (C) where a1, a:3 are allowed to be both zero and T) # O, 1 is given by (4.4.7): * 1 n = u (atk) = 1+Alp(a) + 2P(a)Q(a) 2 2 with p(a) = al—i—Egnand lal2 q(a) = a1a4+ a233 = 0(Ialz) and the energy corresponding to the solution (4.4.15) is given by 1(37 + 3%) hgrd Hx(a) = glaf + 3%) — g(ag + ai) + + (a? + 3%) ° (3134 + a233) = C- In order to put (4.4.15) into a simpler form. we set a1 = woos 9, a3 = wsin 9 a2 = pcos w. a4 = psin w (7,p 2 0) Then, (4.4.15)(a) becomes 74 = _ (1-n)(1-n-UK1), or 3n2 (4.4.16) 4 = [— (1‘")(1‘9‘"*1)]1/4 provided [ ] 2 0 3n2 85 and from (4.4.15)(b),(c), we have s2.) . E-(l-n)(1-n—nA,)]3/4 (4.4.17) p = l;fl_ 3n2 provided [ ] 2 0. Hence, for those values of A, and c satisfying [ ] 2 0, (4.4.16) and '(4.4.17) show that the solution set of (4.4.10) 2-dimensional torus T2 = S1 for 7 to have real positive solutions, we need the condition (4.4.18) — (1‘”)(1‘"'”A1) > 0 3n2 Now, since we can write _ (l-n)(1-n-nk1) _ 1_. 1 ‘ ($93041 3172 _ 3 n 2 . (1:3) the condition (4.4.18) is equivalent to _IL_ (1_n)A, > 1. n x S1 depending on A, and c. Now, forms a in order Hence, if A, > 0, we need-f2; >-—1— i.e., o < 1'" < A, and if A, < o, A,’ 1 then we need 4n— < —l—, i.e. O > 1_n > A1. But, -—» = 1-n K1 n n JIH “1 = Al p(a) + 86 2p(a)q(a). Since 0 < p(a) < 1 and p(a) a 0 as af + a3 a O and q(a) = a1a4+ azaa = 0(laI2), we know that when A, > O. the condition 0 < K,p(a) + 2p(a) q(a) < A, is indeed satisfied for sufficiently small IaI. Similarly, when A, < 0 and Ial sufficiently small, the condition 0 > A1p(a) + 2p(a)°q(a) > A, is satisfied. Therefore, the solution (4.4.16) and (4.4.17) are valid for sufficiently small Ial. To obtain a direct relationship between 7 and p, we eliminate n from (4.4.16) and (4.4.17). Rewrite (4.4.16) and (4.4.17) as 14 = — 1:35L , where k = k(h1.c) = I9; 3k2 p = Ikl'r8 . Also, the energy condition (4.4.10)(5) can be rewritten in terms of polar coordinates as $4 — épz + §A172 + 72(7p cos 9 sin W + 1p sin 9 cos w) = c, or (4.4.18) 72 - p2 + A172 + 273p sin (9 + W) = 20. Now, recall that 1512 (512+k1(5?+53)+2(5?+53)'(5454+5433) n = 11*(5170 = and also recall that the solution 5(h,,c) lies on the energy surface 87 Hk(a) = c and on the momentum surface H2(a) = m. But, since c—m = HNa‘) - 112(5) = 9.546% + (34+ 3345.53 3253). we can write 1512 (4.4.19) n e “*(EIA,) = lal2+ 2(c-m) Thus, we know that if c > m then 0 < n < l and if c < m then n > 1 in a sufficiently small neighborhood of the origin. If c = m then n = 1 and so our system 2 = JVH)‘(z) reduces to the linear system 2 = JVH2(Z). which we have already considered in Section 2. Therefore, we can consider two cases: Case (i): c > m Then 0 < n < 1, so k = 1%,- > 0. Hence, from (4.4.16) and (4.4.17), eliminating k, we have (4.4.20) 37p2 - h1p + 1° = O (7,p 2 0). Thus, when c > m, the critical points of g must lie on the curve (4.4.20) in the (7,p) - plane. Notice that (4.4.20) is quadratic in p and so can be solved for p1 i 2- 4 415?. (4.4.21) p = ‘1 Jki 127 (o < 7 g 12. p > 0). 67 Since p > O, we must have A, > 0. If A, < 0, then (4.4.20) has no positive solution for p and hence the system (4.4.10) has no nontrivial solution. i.e.. has only the trivial solution a = O with c = 0. Notice that if A, = 0, then (4.4.20) has only the trivial solution a = O with c = 0. The graph of (4.4.21) with various values of A, > O is shown in Figure 4.4.1 where 35" =0. < Figure 4.4.1 > Now, the energy surface (4.4.18) can be rewritten as (1 + N,)72 - p2 + 2a7°p = 2c, or (4.4.22) p2 - 2a7°p + 2c - (1+A,)72 = o (A,> o, Ial g 1), where a = sin (9 + w). Also, (4.4.22) is quadratic in p and so can be solved for p: (4.4.23) p = awa : 176276 + (1 + A,)72 - 2c (A,> O). 89 Now, we first consider the case c = 0. Then (4.4.23) becomes (173 i 7 11 + A, + 01274 ‘0 || Since p > O, we must have (4.4.24) p0 = a73 + 411 + A, + a274 (A,> o, Ial g 1). Notice that dpo d7_ 7:0 : Jl+h,-e 1 as A, 4’0+ and po 2 7J1+>x1 for 7 sufficiently small. dp Also, when a > 0, 3:9-> 1 for all A, > o and po z 26173 for 7 >> 1. When a < O, we have po 2 a73 - a73 = O for 7 >2 1. If a = 0, then (4.4.24) becomes p0 = 741+A, . The graphs of (4.4.24) for a > O, a < O, and a = O with O < A, << 1 are shown in Figure 4.4.2. fi d-----..--’.---O------ hn7>°‘ ----o ‘0‘. ‘0 < Figure 4.4.2 2 Hence, from the graphs of (4.4.21) and (4.4.24) it is clear that for sufficiently small values Ial and A, > 0 they have a unique intersection point for any values of la] 3 1, which indicates a torus T2 = S1 x S1 of critical points lying on the 3-dimensional energy surface HA(a) = c = O in the space R‘ = (a,,a2,a3,a4)}. Next, we considere the case IcI ¢ 0 sufficiently small: If c < 0, then from (4.4.23), we have (4.4.25) pc_= a73 + Ja276 + (1+A,)w2 - 2c (0 < o. A,> O,Ia| g 1). 91 dp _ Note that pc_(0) = 42: > o and 3:9 = 0. Also, 7:0 .pc_ 2 p0 and ' pc_ z po for [CI << 1. The graphs of (4.4.25) for a > 0, a = 0, a < 0 with c < 0, A, > 0 are shown in Figure 4.4.3 5’ ' “pool-‘0 0K0 A A - ---.-----—-------pc----- -- < Figure 4.4.3 2 Hence, in the case of c < 0, as in the case of c = 0. clearly the graph of (4.4.25) intersects the graph of (4.4.21) exactly at one point for any values of a and hence we have a torus 'I‘2 = S1 x S1 of solution points in R‘ for the system (4.4.10). Now. we consider the case c > O 1 From (4.4.23), we have 92 (4.4.26) pc+ = a73 i Ja276 + (1+A,)~r2 - 2c (A,> o. c > o, Ial g 1). Notice that the radicand f(7) = (1276 + (1+>\1)72 — 2c is an increasing function of 7 for 7 2 0 with f(0) = -2c < 0 and so f(7) has a unique positive zero 7': 7(N1, c, A) for any A, > 0, c > 0, la] g 1 with the property that 7(7\,, c, a) ->O+ as c ->O+ 7(A1, c, a) decreases as A, or Ial increases. Also notice that even though 7()\,, c, a) can be computed exactly by using the Cardan’s formula, this expression is too complicated to be of any practical use for our purpose. Since pc+ must be pc+ 2 0, we also note that if a g 0, we must have a78 + Ja276 + (1+>\,)72 - 2c (a g 0, A,> 0.0 > 0), (4.4.27) pC+ J(1+7\1)72 - 2c for 7 << 1 (7 2 1%EX-). 1 22 Furthermore, in (4.4.27) with a g 0, since pC+ > 0, the domain of pC+ 20 must be 7 2 J1+A1 and in this domain, pC+ is an increasing funtion of 7. The graph of (4.4.27) with a g 0 for sufficiently small 0, A1, 7 is shown in Figure 4.4.4 together with the graph of (4.4.21) with the various values of A1. 93 '- Pr)” < Figure 4.4.4 ) Hence, in this case a g 0, it is clear from the graph that for each given c > 0 sufficiently small, there is a k0 = Ao(c) > 0 such that if A, ( No, there is no intersection point and if A, = ho. there is one intersection point and if A, > No, there are two intersection points. If a > 0, we have two cases in (4.4.26). (4.4.28) pc+ = a7a : Ja276 + (1+}\,)72 - 2c (0 < a S 1. A,> 0, c > 0) In the + case, i.e (4.4.28)(a) pc+ = a78 + ((1276 + (1+?\,)72 - 2c (a > O, A, > 0. c > 0) 94 x J(1+A1)72 — 20 for 7 << 1, we know that pc+ is defined for 7 2 7(A,,c,a) and pC+ is an increasing function with range pc+(7) 2 a73. In the - case in (4.4.28), i.e., (4.4.28)(b) pc+ = a73 - Ja276 + (1+?x1)72 -20 (a > O, A,> 0, c > 0), we notice that pC+ is.a decreasing function in the domain 2c 7(k,,c,a) g 7 g J1+A1 with a73 2 pC+ 2 0. But, since 7(A,,c,a) x ITEX' for c > O sufficiently small, the graph of 1 (4.4.28)(b) exists in a very small interval and hence the overall graph of (4.4.28) shown in Figure 4.4.5 looks almost like that of (4.4.28)(a). Moreover, we can see that for 7 sufficiently small (178 < 6 for all 0 < A, << 1 7 and so for sufficiently small 0 (i e., sufficiently small 7) the starting point (71a7b) of the graph of (4.4.28)(a) lies below the lower branch of the curve (4.4.21) as shown in Figure 4.4.5. 95 t o to o I."II-....‘ -4, __ _ Y Y Vic/(HA1) < Figure 4.4.5 ) Thus, even in the case a > O. we can still say that for each c > 0 sufficiently small there is a No Ko(c) > 0 such that the graph of (4.4.21) intersects that of (4.4.28) at two points for K > A0, at one point for_h = A0 and at no point for A < A0 Now, we consider the second case: (case ii): c (gm Then, from (4.4.19) n > 1, so k = lgfil< 0. Hence, from (4.4.16) and (4.4.17), eliminating k, we have (4.4.29) 37p2 + A1p + 7° = O (7.p 2 0). or -i 2- 4 (4.4 30) p = N17 JR, 127 (A,< 0). 67 ' 96 Note that since p > 0, we must have A, < 0. If A, 2 0, then (4.4.29) and hence our system (4.4.10) has only the trivial solution a = O with c = 0. The graph of (4.4.30) with A, < 0 is the same as that of (4.4.21) with A, > 0. Now, the energy surface (4.4.23) becomes (4.4.31) p = a73 : Ja276 — 2c + (1+k,)72 (A,< o. Ial g 1). If c = 0, then (4.4.31) becomes p = a73 i Ja27 + (1+A)72 Since p > 0, we must have (4.4.32) p = a73 + Ja276+ (1+A)w2 (A,< o,|A,|<< 1, Ial << 1). Also, we notice that 35- = 41 + A, < 1 and p 2 7°41+>x1 for 7 << 1. I‘=O The graph of (4.4.32) is almost the same as that of Figure 4.4.2 as long as IA,I << 1 and '7' << 1. Also, in the case of c ¢ 0, the graphs of (4.4.33) pc_= a73 + Ja27 + (1+A,)i2 - 2c (-1 << 0 < o, -1 << A, < o, lal s 1) (4.4.34) pc+= a73 :_Ja27 + (1+A,)72 - 2c (0 < c << 1, —1 << A, < 0, la] << 1) 97 are almost the same as those of Figure 4.4.3 and Figure 4.4.5 for IA1I << 1. Therefore, from the above analysis, we can state the following conclusion: Theorem 4.4.1 Consider the Hamiltonian system (4.4.35) 2 = JvH*(z) with HNzl = g (xi+ y?) — 5 (x4 + ya) + A.(x=;= + yi’) + (xi + yixxor. + xzyi) in the normal form with respect to H2(z) = g(xf + yf) — g(xg + yg). Let HA(z) = c and H2(z) = m. Then, in a sufficiently small neighborhood of the origin and for sufficiently small Ih1I and IcI, we have the following: (i) when c > m and c g 0, the system 2 = JVHA(Z) undergoes 93 I/\ supercritical bifurcation from the equilibrium solution 2 = 0 for A, 0 when c = 0 and from ¢ for A, < 0 and S1: x3 + y3 2 -2c for A, = 0 when 0 < O to a continuous family of periodic solutions of the form *— eJAt/ u (a’xi)a(k,,c) (4.4.36) z(t) = for each small A, > 0, lying on a torus with period 2wu* < 2w. (ii) when 0 > m, and c > O sufficiently small, there is a A0 = ho(c) > 0 sufficiently small such that if A < A0, then the system (4.4.35) has no periodic solution and if A = NO. then (4.4.35) has one continuous family of periodic solutions of the form (4.4.36) lying on the torus with period 2rn* < 2r and if Ao< A << 1, then (4.4.35) has two disjoint continuous families of periodic solutions of the form (4.4.36), each 98 lying on the corresponding torus with corresponding period 2wu* < 2w. (iii) when c < m, we have the same kind of bifurcaiton as in (i) and (ii) except that in the case of (i), (4.4.35) undergoes a subcritical bifurcation from 2 = 0 for A, 2 0 to a torus for A, < 0 and in the case of (ii), from no periodic solution for A0 < A, < 0 to a torus for A, = A0 and to two disjoint tori for —1 << A, < Ao < 0 with corresponding period 2wu* 2 2w. §5. Local bifurcations of periodic orbits as the eigenvalues split across the imaginary 8X18. Now, in this case our Hamiltonian Hx(z) takes the form (4.5.1) H>‘(z) = % zTAz + %2T13(A)z + fi,(z), where A = diag (1, -1, 1, -1), l 0 A.‘ I A. o A, o | O H;(z) = (X? + YT) ° (X1Y2 + X2Y1)- The corresponding Hamiltonian system becomes 99 (4.5.2) 2 = JVHA(z) = JAz + JB(A)z + Jvfig(z). 2 Recall that u*(a,A) = lal Ial2 + < Aa, B(A)a + VH;(a) > Computing the right—hand side, T A3 = (31- "azo a3, ‘34) . T B(A)a = >\lrli'flmasvaziai) - < Aa, B(A)a > = 0 for all A,. * Hence, in this case p (a.A) becomes (4.5.3) “*(a,A) = Ia)2 Ialz + 2(37 + a5)(aia4+ a233) 1 = 1 + 2p(a)°Q(a) 2 2 where p(a) = 21—i—293 O < p(a) < 1 lal2 q(a) = (3134 + azaa) Also, the funtion g(a,A,n) becomes g(a.A.n) = H2(a> - nH‘(a) =mm)-nmo)+®e)+mmn 100 (140423 Aa - 17% 4T B(A)a - Trida) 1 = (l-n)§(a¥ ’ 35 + 35 ‘ 33) ‘ W°Aq(aia4+ a233) -n(af + a3) ° (a1a4+ a233)- Hence, our gradiant system Vg(a,A,n) = 0 becomes p- (1-n)ai ‘ ”A434 — n [34(37 + 35)+ 231(aia4+ 3233)] = O (1) (4.5.4) , ‘(l-nlaz ‘ ”A433 ‘ W'as(37 + 35) = O (2) (l-n)ao ‘ "A432 ‘ n[a2(af + 35) + 233 (3134+ a2a3)] = O (3) t “(l-n)a4 - "A431 ‘ "31(a7 + 35) = O (4) together with the energy HA 1 2 2 1 2 2 2 2 (a) = filai + as) — §'(az + a4) + A4(a,a4+ a23:3) + (a1 + a0)(a1a4+ azaa) = c. (5) 1 1+2p(a)Q(a) explicitly. Hence, if q(a) = a,a4 + azaa > 0, then 0 < n < 1 and if Note that in this case n = u¥(a,A) = does not depend on A q(a) < 0, then n > 1, and if q(a) = 0 then n = 1. Also system (4.6.4) has the trivial solution a = 0 for all A4 with energy 0 = 0 and still has the reflection symmetry a,ee'a3, azee’a4. In the following, we consider several cases: case (i): n = 1 (i.e., (af+a§)(a,a4+ azaa) = 0). Since a? + a3 = 0 i.e., a, = a3 = 0, is a.special case of a,a4+ azaa = 0, we may only consider the general case a,a,+ aeaa = (L In this case, the system 101 (4.5.4) reduces to ( -A4’a4 = adaf + a?) ' (1) (4.5.5) , ‘9‘an = aslaf + a:23) (2) ‘4432 = 32(37 + 35) (3) . —A4ai = 31(37 + 35) (4) together with Hk(a) = é(af+ a3) — $(a§+ afi) = c (5) If A4 =v0, then (4.5.5) has the solution.a, = a3 = 0, a2 and a, are arbitrary values with the energy HR = - gag + a3) = 0. Hence, when q(a) : 0 and A4 = 0, the solution set of (4.5 5) is 81 4 1 2 2 = (a 6 R I a, = a3 = 0, - i(a2 + a4) = c} for each c < O a = 0 for c = 0 o for c > O . Therefore, when q(a) = 0 and A, = 0, our Hamiltonian system (4.5.2) has the following 27 periodic solutions! 2 (exp JAt) a with a = (O,a2,0,a4),- g(az + afi): c foreach (4.5.6) z(t): o for c = o C < 0 o for c > 0 Bgmggk: In the case of A4 = O and q(a) = O, the system (4.5 2) reduces to the linear system 2 = JAz. 'The above result (4.5.6) agrees completely with that of global analysis given in Section 2. (See Figure 4.2.2). If A, = O and q(a) = 0, we have 9 = —h (in the notation of Section 2)(critical values of H x L) and so 102 _1 {0) x S1 for h < 0 (H X L) (hr-h) = (0) for h = o. o for h > o . If A, ¢ 0 (and q(a) = 0), then from (4.5.5) (2),(4) we have H O as(>\4 + a? + a3) = O (4'5'7) a,(A, + a? + a3) . subcase 12 a, = a3 = 0 Then from (4.6.5) (1),(3), we have only the trivial solution a = 0 for all A, with energy 0 = 0. subcase 21 a, ¢ 0 or a3 ¢ 0. (and A, ¢ 0) Then from (4.5.7), we have A, + a? + a3 = 0. If A, > 0, we have no solutions a,,a3. If A, < 0, we have solution set for a,,a3 of the form af + a3 = -A, which forms a circle in the (a,,a3)-plane with radins VCAIZ Corresponding: to these 'values of a,,a3. we can obtain. a2,a, (4.5.5) (1) ,(3) which become "A434 = a, ° ("K4) -A,a2 = a2 ° (—A4) and hence a2,a, may be arbitrary satisfying the energy condition HNa) = 5—H.) — go; + at) = c from 103 or a3 + af = -2c - A, which form a circle on the (a2,a,)-plane if c < - %1(A, < 0). and a2 = a, = 0 if c = - gin For A, > 0 or for A, < 0 and c > - g4. solutions for (4.5.5). Thus, we have the following conclusion. we have no Theorem 4.5.12 In the case of q(a) = 0 (i.e. linear system) the system (4.5.2) has the following 2w-periodic solutions as A, varies for each energy level 0. (i) when A, = 0 eJAt a with initial points a = (0,a2,0,a,) lying z(t) = on the circle — é-(ag + a3) = c for each c < 0 O for: 0 o for c) 0 . (ii) when A, < 0 r JAt . . . . . . e a With initial pOints a = (a,,a2,a3,a,) lying on torus a? + a3 = —A, and a3 + afi = —2 c —A, z(t) = < if -A, > 20 eJAt a with a2 = a, = 0 and a? + a3 = -A, if —A, 2 2c ~¢ 1f-A,<20. (iii) when A, > 0, we have no periodic solutions except 2 = 0 with energy 0 = 0 In other words, in the case of linear system, (a) For energy level 0 < 0, the system (4.5.2) undergoes a subcritical 104 bifurcation from (0} x S1 to T2 as A, varies from 0 to A, < O. (b) For c = 0, (4.5.2) undergoes a subcritical bifurcation from {0) to T2 as A, varies from 0 to A, < 0. (c) For c > 0, (4.5.2) undergoes a saddle-node type subcritical bifurcation at A, = —2c from ¢ to T2 via S1 as A, varies from 0 to A, < —2c via A, = -2c . /// Next, we consider the genuine nonlinear case: case (ii): n g 1 (i.e.. q(a) = a,a, + azaa ¢ 0) In this case, we note that 0 < n < 1 if q(a) > 0 and n > 1 for q(a) < 0. Also note that (4.5.4) still has trivial solution a = O for any A, with energy c = 0. From the system (4.5.4) (2), (4), we have — 1-" a3(k4 + a? + 8.3) (4.5.8) n 1-n a,(A, + af + a3) . Substituting (4.5.8) into (4.5.4) (1), (3) we have, by the refletion symmetry, ai[(1-‘n)2 + 7720M + a? + 33) ( M+ 3(8‘133“ 83”] = 0 (4.5.9) 33°[(1‘n)2 + ”2(k4 + a? + 35)(A4 + 3(af + 35))] = 0 Thus, the 4 x 4 system (4.5.4) has been reduced to 2 x 2 system (4.5.9). Now, notice that because of the assumption (af + a§)(a,a, + a2a3) ¢ 0 a, and a3 cannot be both zero in (4.5.9). Hence, (4.5.9) reduces to 105 the equivalent equation (4-5-10) (l-n)2 + ”2(A4 + a? + 35)(A4 + 3(37 + 35)) = 0- Also notice that if n = 1 then (4.5.10) includes the case A,+ a? + a3 = 0, in which case our system (4.5.4) reduces to the linear case we have already considered. Let X = af+ a3 2 0 and Y = a§+ a3 2 0, then (4.5 10) becomes _ 2 (4.5.11) 3X2 + 4A,- X + AE + [$53] = 0. Also, from (4.5.8), we have (4.5.12) Y = [fir—{r x - (x + A,)?-. Eliminating lgfl-from (4.5.11) and (4.5.12), we have _ —X(X+A.)2 _ —X(X+A,) _ (3X+A,)(X+A,) _ 3X + >\4 (4.5.13) Y (X,Y > 0) Here we assumed that X + A, g 0 for if X + A, = 0 then from (4.5.11) we have n = 1 and so our system reduces to the linear case. Notice that if A4 := 0, (4.5.11) and (4.5.12) has no solution except the trivial solution a = 0 with c = 0. Furthermore, since X,Y 2 0, from (4.5 11) and (4.5.13), we must have (4.5.14) A, <0and-g‘—* 107 Now, the energy surface (4.5.4)(5) can be rewritten as 1 2 1 2 2 . i7 — 5p + (7 + A,) 7p Sin(9 + W) = c, or (4.5.16) p2 — 2a . 4(42 + A,)p + 2c - 72 = o (A,< 0, [alg 1), where a = sin(6 + w). Since (4.5.16) is quadratic in p, we can solve it for p1 (4.5.17) p = a7(72 + A,) : Ja272(72+A,)2 + 72 -2c (A, < O, Ial g 1.) First, we consider the case 0 = 0: Then (4.5.17) becomes (4.5.18) p0 = a7(72 + A,) + 7 - Ja2(72 + A,) + 1 (A, < O, Ial g 1) since p0 > O. 22 Notice that for A, < o and IA,I << 1 and [7| << 1, p0 dp ————————— 3-2- : aA, + JazA,+ 1 (A,< O), 7 7:0 dpo and so a——- > 1 for a < 0, = 1 for a = 0, 1 for a < 0. 7 7:0 Note that if a = 0 then a,a,+ azaa = 0 and hence our system reduces to the linear case. 108 The graph of (4.5.18) for a > o. a = o, a < 1 with -1 << A, < o is Shown in Figure 4.5.2. C. 00° d- o 0< 0 7:0 "(=0 for a < O, = 0 for a = 0, < O for a > 0. Also. pc_ > p0 and pc_ 2 p0 for 7 2) Icl. The graph of (4.5.19) for a > 0. a = O, a < 0 with c < 0. A, < 0 is shown in Figure 4.5.3 9 -.-------“----au-ooo--- < Figure 4.5.3 2 Hence, in the case of c < 0, as in the case of c = 0, clearly the graphs of (4.5.19) and (4.5.15) intersect exactly at one point, which corresponds to a torus T2 = S1 x S1 of critical points of g lying on the energy surface HA(a) = c < 0 in R‘. Now, we consider the case 0 > 0: From (4.5.17), we have (4.5.20) pc+ = a7(72 + A,) : Ja272(72 + A,) + 72 -2c (0 > O. A, < 0. lal $1). 110 Let f(7) = (1272(72 + A,)2 + 72 - 20. Then f(0) = -2c < 0 and f'(7) = 27[a2(72 + A,)(372 + A,) + 1]. Hence f'(7) > 0 for ngi-g 7 g J-A, << 1, i.e., f'(7) > o for all 4 > o if A, < o, IA,| << 1. Thus, f(7) has a unique positive zero 7(A,,c,a) with 7(A,,c.a) 470+ as c-é 0+. Moreover, since f(J-A,) = -A,- 2c, if — A, = 20 << 1 then 7': J-A, and so (4.5.15) and (4.5.21) have one intersection point (J-A,. 0). If —A, 2 20, then 7'< J-A, so they have one intersection point T2 = S1 x Si. If -A, < 20, then 7 > J—A, so they have no intersection point. Since J_34 g 7 g J—A, in (4.5.15), we may only consider pc+ for 7 _<_ J—A,. So, if we assume 7 g J—A, in (4.5.20), then for a 2 0, (4.5.20) must be (4.5.21) pc+= a7(72 + A,) + Ja272(72+ A,)2 + 72 - 20 (a 2 0, c > O. A, < 0). Since pc+(7) g 0 and pC+(J25) = 0, pC+ must have the domain 7 2 420. Also, pc+ x 442 — 2c for r << 1 and |A,| << 1. 111 The graph of (4.5.21) is shown in Figure 4.5.4. P . - - ._ J. -221 1’ an; < Figure 4.5.4 2 Thus, it is also clear from the graphs of (4.5.15) and (4.5.21) that for given c > 0 sufficiently small, they have one intersection point T2 = S1 x S1 if — A, 2 2c and one intersection point S1 on the 7-plane if -A, = 2c and no intersection point if -A, < 2c. For a < 0, we have from (4.5.20) 4.5.22 p = a7 72 + A, + Ja272 72 + A, 2 + 72 - 20 c+ — (a < o, A,< o, c > 0). In the + case, we have (4.5.22)(a) p:+ = a7(72 + A,) + Jaz72(72 + A,)2 + 72 - 2c A,< o. c > 0) z 412 - 2c for 7 << 1, IA,I << 1. (a < 0, 112 But, p:+(?) = a7(72 + A,) 2 o if 7 g J-A,, i.e., -A, 2 20. That is, if —A, > 20, then I? < J—A, so p+ if -A, = 20, then 7': J-A, = 420 so p:+(7) = 0. if -A, < 20, then 3‘ > J—A, so p:+(:) < o and hence p:+ is defined for 7 2 J25: Also, in the - case, we have (4.5.22)(b) p;+ = (17(72 + A,) — Ja272(72 + A,) + 72 -2c (a < O. A,<0,c>0). Similarly, we have (i) if -A, > 20, then 7'< J-A, so p;+(7) = p:+(7) > O and p;+ (320) = (A Hence in the case —A, < 2c, pC+ is a decreasing function defined on the extremely small interval [71 320] with 12c < J-A, and the combined graph of (4.5.22)(a) and (b) in this case looks like the one in Figure 4.5.5.(a). (ii) if —A, = 2c, then 7': J—A, = J20 so p;+(7) : 0 and moreover for 7 > 7': J-A, = 42c, pc+ is not defined, in other words, the graph of pC+ is just one point (42c, O) in this case. Hence, in the case —A, = 2c, the combined graph of (4.5.22)(a) and (b) looks like the one in Figure 4.5.5(b). (iii) if -A, < 2c, then 7') J-A, so p;+(7) < 0 and hence p;+ must be 113 defined for 7 2 425 but for 7 > 425, p;+ < O, that is. p;+ is defined just at one point (425; 0). The combined graph of (4.5.22)(a) and (b) is shown in Figure 4.5.5(c). 9 (0L) «A,>2c P . (b) -A,=2t ’ (m. 2 r; * 2,. P t (C) -A4,<2-C + E - 94a 2 - -Y J-‘A, 72? < Figure 4.5.5 2 Therefore, when c > 0, even in the case of a < 0, we can still say the 114 same thing as in the case of a 2 0, that is, that if -A, 2 2c then we have a torus T2 = S1 x S1 of critical points, if -A, = 2c then we have a circle 81 of critical points on 7-plane, if —A, < 2c then we have no solutions for our system (4.5.4) on the energy surface Hk(a) = c > 0. Thus, we can state the following general conclusions including the linear case. Theorem 4.5.2: Consider the Hamiltonian system ° A (4.5.23) 2 = JVH (Z) . A 1 1 Wlth H (Z) = §(X‘f + y?) - §(X§ + 3'3) + >‘4(X1YZ + Xzyi) + (X? + y?)° (x,y2 + xzy,) in the normal form with respect to H2(z) = éfxf+ y?) - é(x§+ yg). Let HA(z) = c. Then, in a sufficiently small neighborhood' of the origin and for sufficiently small IA,I and IcI, we have the following: (i) when c = O; the system (4.5.23) undergoes a subcritical bifurcation from 2 = O for A, 2 O to a continuous family of periodic solution of the form x. _ (4.5.24) z(t) = eJAt/u (21'7“) a for A, < 0 lying on a torus T2 = S1 x Si. (ii) when c < 0; the system (4.5.23) undergoes a subcritical bifurcation from ¢ for A, §>0 and from S1 on (x2,y2) — plane for A, = O to a continuous family of periodic solutions of the form (4.5.24) for A, < 0, lying on a torus T2. 115 (iii) when 0 > 0, then for A, 2 -2c the system (4.5.23) has no periodic solutions and for A, = -2c the system (4.5.23) has a periodic solution S13 x? + yf = 2c lying on the (x,,y,)-plane and for A, < -20 it has a continuous family of periodic solutions of the form (4.5.24) .lying on a torus T2 = S1 x S1 in R“. Furthermore, in each case (i), (ii). (iii), p* < 1, = 1, > 1 depending on x,y2 + xzy, > 0, = 0, < 0 respectively. Remark: So far we have considered a truncated Hamiltonian containing only one fourth order term. However our methodology can still be extended to the case containing the whole nine fourth order term and can even be extended to a nearby nonintegrable system by combining Moser-Weinstein reduction. 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