WW I‘ N ”NH/W!MLVIN/HUI“! H H WW! WL _x _\ 01 m '—4 18 IUD-h IIII IIII IIIIII IIIII I II II 3 1293 00839|5 LIBRARY Michigan State University This is to certify that the dissertation entitled . BIFURCATING PERIODIC SOLUTIONS OF A SMOOTHED PIECENISE LINEAR DELAY DIFFERENTIAL EQUATION presented by Derek Graham Lane has been accepted towards fulfillment of the requirements for PhD Mathematics degree in {L/NI/ (/I‘LJ Major professor Date March I, 1988 MS U is an Affirmative Action/Equal Opportunity Institution 0-12771 MSU LIBRARIES RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. BIFURCATING PERIODIC SOLUTIONS OF A SMOOTHED PIECEWISE LINEAR DELAY DIFFERENTIAL EQUATION By Derek Graham Lane A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements of the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1988 '7‘ $76‘ .97 2'" I .1.“ ABSTRACT BIFURCATING PERIODIC SOLUTIONS OF A SMOOTHED PIECEWISE LINEAR DELAY DIFFERENTIAL EQUATION By Derek Lane The neighbourhood ofa periodic solution Ofa delay differential equation :c'(t) = af(:z:(t — 1)) is studied. The degree of the periodic solution is shown to change at a large number of parameter values. Bifurcation of periodic orbits follows as a result, for arbitrarily large values of a. Choosing f nearly piecewise linear simplifies formal calculations of a where the solution has neutral stability. The degree-Of the periodic orbit is studied through the spectrum of the linearization of the flow around the periodic orbit. Using ideas of Walther this is reduced to a set of ordinary differential equations with boundary conditions. A two part argument based on the special choice of f gives the approximate positions of a where the periodic solution can change its degree. The ordinary differential equation is studied by estimating the effect of smoothing, and calculations of the motion of a critical eigenvalue are obtained by matching arguments for maps related to the ordinary differential equation. The calculations are carried out by hand and checked symbolically by computer. ACKNOWLEDGEMENTS I would like to acknowledge the assistance and patience of my advisor Professor Shui Nee Chow over the prolonged preparation of this work. My thanks go also to Professor John Mallet-Paret for his aid in exploring differential equations, and to Professor Hans Otto Walther for encouragement and suggestions. I am grateful to Professors Sheldon Axler, Charles Seebeck Jr., and Clifford \Veil for their patient reading of my thesis and attending my defence. I wish to thank Ms. Margaret Memory for proofing an early draft of the thesis, and Ms. Barbara Miller, the secretary of the graduate office of the Department of Mathematics for her help with things organizational. ii TABLE OF CONTENTS Chapter INTRODUCTION 1 FINDING BIFURCATION PARAMETERS FOR z’(t) = af(:1:(t -1)) . . . 1.1 Introduction 1.2 Definition of Flow and Solutions 1.3 Construction and Study of a Periodic Solution 1.4 Linearization Around the Periodic Solution 1.5 Satisfaction of a Necessary Condition 1.6 Determination of Bifurcation Parameters 2 CIIANGE OF DEGREE FOR W) = emu — 1)) 2.1 Introduction 2.2 Definition and Discussion ofa Bifurcation Function 2.3 Technical Preliminaries 2.1 Calculation of Properties of the BifurcationFunction 3 CALCULATIONS 3.1 Computation of 539-9: 3.2 Calculation of 2—2 3.3 Machine Calculations LIST OF REFERENCES iii Page 19 23 27 27 38 29 41 ~11 44 49 Figure 1 FTgure 2 . Figure 3 . FWgure‘I . LIST OF FIGURES Page 19 33 INTRODUCTION The equation 7 99(1) = gI-"CIt -1)) (0-1) with a continuous function g : R —+ R satisfying g(O) = 0 and g'(0) < 0 has been studied as a model of physiological control processes. For‘this application 'see Mackey and Glass [5], for other applications see, for example, the references in Walther [11]. Numerical simulation of shows that there are useful choices of g with complex behaviour. An enlightening way ofstudying equation (0.1) is to choose g = of where f satisfies f’(0) = —1 and a varies over the positive reals. There is a Hopf bifurcation at a = 7r/2, giving rise to a continuum of periodic solutions of equation (0.1); with extra conditions on f these solutions are stable for a close to 7r/2, and there are conditions on f which imply global stability among an important class of solutions. Other approaches to the study of equation (0.1) exist. One may choose f carefully to simplify the study of solutions to equation (0.1). Choosing f as an odd function allows the discussion of special period 4 solutions branching from the Hopf point as solutions of an ordinary differential equation with boundary conditions. see Kaplan and Yorke [4]. One may study the equations for large a, searching for fix points of cone maps using Schauder’s fix point Theorem, as in Nussbaum [10]. Finally, one may choose f so simple that all solutions may be explicitly calculated, hoping for complex behaviour even with a choice of f close to a step function, see, for example, Walther [12]. \\'e have chosen an odd function which is nearly piecewise linear to gain insight into a neighbourhood of the bifurcating periodic orbit that can be tracked using 1 2 techniques from Kaplan and Yorke [I]. For large 0: we obtain many new periodic solutions by a degree argument, these solutions being very different from those found by Nussbaum [10]. The choice of f is tailored to simplify the ordinary differential equations Walther uses to study the degree of the special period 4 solution. A difficulty with this choice of f is the smoothing necessary for the applicabil— ity of the standard theory, which requires smooth f for local analysis around the periodic orbit. All of the arguments are affected by this problem, some more than others. Chapter 1 is devoted to setting up enough estimates on smoothing’s effects to get possible linearized neutral stability for the periodic solution. In Section 3 we show that some techniques from \Valther [11] give us a unique periodic solution satisfying the equations of Yorke [4]. we also find estimates of the size of various ies quantities needed later for the study of change of degree. Section 4 introduces a natural recasting of the problem in terms of simple maps, and a simple argument available in this case provides an alternative to part of \Valther’s analysis of the problem. In Sections 5 and 6 enough properties of the maps are derived to find many a’s where the periodic solution could have linearized neutral stability. \Ve use the special choice of f heavily here. In particular, a fortunate property of f makes the search for points of possible neutral stability Split into two parts, simplifying the search noticeably. This part of the thesis is much less affected by smoothing f than is Chapter 2. Chapter 2 is devoted to showing that the degree of the periodic orbit does indeed change near the 0’s found in Chapter 1; it is here that the main technical difficulties arise. The first problem, solved in Section 4, is to find an expression for the contribution of the smoothing to the rate of change of an appropriate eigenvalue. The second problem, solved for technical reasons in Section 3, is to find a key quantity in the calculation of Section 4. This requires a matching argument in the three natural maps defined in Chapter 1, Section 4, and estimates from Chapter 1 to justify the matching. In Section 4 of Chapter 2 the main Theorem of the thesis is proved, quoting Chapter 3, the straightforward calculations needed to finish the proof. The last few pages of Chapter 3 are a record of a symbolic calculation of the relevant quantities by machine. This was done to check on hand calculations. Nu— meric calculations are restricted in application for large a’s since quantities appear in the analyses of exponentially small order in a. Chapter 1 FINDING BIFURCATION PARAMETERS FOR :c’(t) = af(x(t — 1)) 1. Introduction \Valther [11] has studied a class of equations of the form x'Ii) = MIN-1)) (1-1) and has shown that a periodic solution of this equation undergoes at least one bifurcation as a varies. For particular choices offclose to a piecewise linear function which almost satisfy the requirements of \Valther’s theorems we are able to show the periodic solution undergoes a large number of bifurcations as a varies. In the next four sections we introduce the notation necessary to study the periodic solution and to study its stability. The major objects of interest are the periodic solution :r"(t), and a Poincare map P with an associated degree found by a study of equations (4.3) and (4.4). These equations are a two—parameter two-dimensional linear time dependent set of ordinary differential equations with boundary conditions. The notation and framework of study in these sections follows \Valther[11], which uses ideas of Kaplan and Yorke [4] to find the periodic solutions, and special properties of the solution to simplify the study of its stability. Our choice of f further simplifies the analysis of stability enough so that in sections 5 and 6 we are able to find choices of a where 1:“ has linearized neutral stability. Change of stability and hence bifurcation is studied in chapter 2. 2. Definition of Flow and Solutions TVe consider a one-parameter family of delay differential equations: mix) = area — 1)) (2.1) where a, :2: E R and a, the parameter, is chosen positive. We would like to study this family with f equal to the piecewise linear function f* whose graph is given in Figure 1. This f“ is simple enough so that formal calculations related to linearized stability of a periodic solution are tractable. IAY ‘ Figure 1 Unfortunately as f‘ is not C1 we know of no linearized stability theory for the flow given by equation (2.1). We approximate f‘ by fa, a C1 function to which the theory in, cg. Hale [3] is directly applicable. We were guided by the formal computations for f‘ throughout our work, however. fa’s graph is given in Figure 2. Figure 2 For definiteness we define fly by: {—13, lf[$[ 0. Because rf(3:) g 0 for I 6 (—2,2) the slowly oscillating solutions defined next have particular importance. Kaplan and Yorke [4], Chow, Diekmann and Mallet- 7 Paret[1], Chow and Walther[2], Mallet-Paret[5], Nussbaum[10]and others have stud- ied this class of solutions in other contexts. Definition 1.A slowly oscillating solution a: is a solution of equation (2.1) where if z(tl) : :r(tg) = O and 11 at t2, then [11 — 12] > I. 3. Construction and Study of a Periodic Solution In this section we show that equation (1.1) has a one-parameter family of periodic solutions, one for each 01. we introduce and study the linearized stability of these periodic solutions as a varies. “"e do this by constructing P, a Poincare map as in Walther[11], p.272. Let a be given. We search for solutions a: of equation (1.1) satisfying :r(t —- 2) = —:r(t). (3.1) These solutions, if they exist, are periodic of period 4. Defining y(l) : a:(t-1) we find that ifa: is a solution of equation (1.1) satisfying equation (3.1) then it also satisfies: 3'0) = afIL/(O) (3.2) 3/0) = -af($(l)), with boundary conditions 95(0) = 31(1)a (3.3) See \Valther[11] and his references for proofs and history of this method. The periodic solutions of the delay equation(1.1) satisfying the symmetry defined by equation (1) are shown by Walther[ll] pp.277—278 to also satisfy equation (3.2) and equation (3.3). Conversely, in a class of functions close to f, Walther has shown that any solution of equation (3.2) and equation (3.3) is a periodic solution of equation (1.1). Kaplan and Yorke[4] show with slightly different hypotheses on f that there is at most one solution of equations (3.2) and (3.3) satisfying y(0) = 0 and 2 > 3(0) > 0. We next modify their existence proofs for our case, starting with some notation and lemmas. Let 7] be any element of B. Let 23(1) 2 [31(1)] and :E(a, 77, t) be :E(t) where :c and y satisfy equations (3.2) and (3.4) below: fim=2-m MW=0- Lemma 1.For all (1, 77,1 i(a, 17,1) 2 :E(1, 7;, at) Proof: :E(1, t], at) satisfies equation (3.2) and has the same initial values as :I:(a, 17, t). Since f is Lipshitz, these solutions must be identical. I We would like to show that 17 determined by the condition :1:(a,17,1) = O :r(a,n,t) > 0 if t6 (0,1) (3.5) is a function of a, and to bound 37% away from zero independently ofa". \\"e do this in the next set of lemmas. We define T = {:5 : :1: + y g 1,:r,y 2 O}. For a: E T we define 6(i) by tan (9(:E) = 3;. \\"e give a result implying monotonicity of 77 as a function of a : Theorem 2.Let f(;z:) be an odd, continuous function such that xf(;r) < 0 on [0,2] and f(:r) = —:r for [at] S b. Assume f(;1:):1:‘1 is strictly monotonic increasing for :1: > I. If :21 and :E2 satisfy equation (3.2), 231(11) : r:i:2(tl) for some r > 1 and 5:,(ll)cT, then WEUDSQQAW for all i > t1 such that 52,-(1) E T. ProofiNussbaum’s proof in [10], pp.27_—29 is applicable. I Corollary 3.x(a,t]1,1) = a:(a,1]2,1) = 0 implies 771 = 112 if 772 and 771 are greater than 1 + 0' and the :E’s remain in T for le[0,1]. Prooszake m > 1+ 0, and assume 772 > 171. Theorem 2’s hypotheses on f are satisfied by fa with b: 1 — a. By explicit calculation 9(;i:(a, m, 6)) > 9(:E(a', 112, 6)) 9 for 6 small enough. Let 171(t) = :E(oz,7]1,t) and 5:2(t) = :i:(a,t}2,t + A!) where At > 0 is chosen so that 5:1(6) = 5:2(6). Theorem 2 now applies to 51,52 (choose t1 = e ). Therefore 0(fé(a,772,1+At)) S %.I[ence€(5:(a,772,1+At)) S g-ver2 Hence x(a, 771, 1) > 0, a contradiction. I Corollary {LEquation (3.5) defines 17 as a function of a if 77 > 1 + a. Prooszhere must be at least one 17 which satisfies z(a,n,1) = 0 since by explicit calculation for all a Z 125, 6(:E(a,1,1)) > 12‘- and lim,,_.00(:2(a,1], 1)) = 0. Continuity of 0 outside a. neighborhood of the origin implies the existence of a choice 770 such that 0(a?(a, 770, 1)) = 12'- . The infinum over all such strictly positive 11’s will be defined to be 77(0). This candidate satisfies Corollary 3 by minimality and hence is unique. It satisfies equation (3.5) by its construction. I We next bound 3% from above: Lemma 5.For all a > g as: l—(as r1, 1)| < 2620’- (3-5) (917 ' Prooszhe variational equation for the vector 3% is: (93:64 0 f;(ra(t))]5_f an LXI/"(1)) 0 (977 where 85: _ 1 RID—lo] Since 0 f;(a=°‘(t)) '“Ifgtyatm 0 I'd“ for all 17, a, t, Gronwall’s inequality implies a— giw < e?“ +1 < 282°". ’7 Therefore 10 Since (91' 83?: we have 0:1: 20, lan(a,n,1)|<26 -- \\"e bound [3%] from below next. Let am 2 inf¢>0{t : :r(a,t}(a),t) 2 1+ 0}, am exists and is unique for each choice of 77 > 1 if a is chosen small (depending on a ). We introduce a function K : R —> R a map bounded on compact sets of R as a notational convenience for this lemma and later results. It will be clear that at least one such function exists making all conclusions of the lemmas true. Lemma 6.Fix 010 > 3. There is a 0'0 2 00(00) such that for all a > 010 and 0<00 (9:1: ['a—(Cl,T], 1)] > |K(ao)af(n) cosh(a'a10,)]. (3.7) 77 Prooszy explicit calculation 3—:(a, mam) = —cosh(aa..> (3.8) @(a, 777011;) = sinh(aala). an Here we used piecewise linearity of fa twice; once to Show the flow stays in the region where f is linear fort < am, and once to calculate 1(a, 0,010,). Choosing (70 small enough so that am > 0 we find 21_ 3—17 69—338,, 313,, 2 sec 9—— — (91) 1:2 Therefore 21L_ 3—1 33—0 = 00529—36" ya" (91] 1:2 8_r_ 2 Z —yan cos 9 11 Since a > ao implies y(a,t},ala) > [C(00) , [:13] < 1 and c0526 > k(a0) for some k(a0) > 0 we have, using equation (3.8), 3-3 _>_ K(a0)cosh(aam). (3.9) Choose to E R depending on A17 > 0 so that QIEIO, 7) — A777110)): 905(0) 17,0101) (3-10) “"e have, using equation (3.9) K [to — am] 2 (a0)_\17 cosh(aa1a). (3.11) oz Defining ya) = :i:(a, n — Amt +10 — 01a), we note that the hypotheses of Theorem 2 are satisfied with 5:1 2 :E(a,17,t) and :13? = 11*. Therefore 9017(0) S 905(0) for l g 1. Therefore 1*(1)=x(a,n—_\n,1+lo—ala)20 (3.12) since at t = 1 9(;E(t)) : 7'. Now for some X 6 [1,1 + to — am] the Mean Value M| Theorem implies 3(a) n - An, 1 +10 — am) - 210,11 - $17.1) = af(y(a, 77 - An. 1000 - am) 01‘ $(a, 1) - A7711) = IIa, 17 - An,1+to — am) - af(y(a, 77 - An, x)(to — cue)- (3-13) Equations (3.12) and (3.13) together imply (to — 01a) To —A 1—xa' 1 I ’77 "A; ( ’0’ )2-af(y(a,n-An,x))*—jn——- (3“) "m u _. ' a‘“. 12 Taking the limit as A1] —> 0 of (14) and using (11) we find 8:1: K a a—n(ain:1) S af(y(ain71) ( 0) a cosh(aa10,). Therefore l%:-(a,n,1)l> K(ao)|f(n)lcosh(aa1a)- - Lemma 7.For all 01 > 3 Proof: 0.7: 0:1: , —(a7 771 1) 2 _(117710) : 2? (1:77:01) (3a 601 = af(y(1, 7), a) = af(y(a,n,1))= aft? - 77) = af(77)-- Putting these lemmas together we next have a bound on 3—3. Note that this bound is independent of 17. Proposition 8.For all a > 3 and a“ chosen as in lemmas 6 and 7, 0(a) is well-defined and a Iaf(n)| S [6—2] g [K(a0)acosh(aa1a)-1|. (3.16) Prooszhe inequality follows from Lemmas 5, 6, and 7 and the relation 3—; = ) , a consequence of the Implicit Function Theorem. I QJQJ QH ( QJIQJ :1 H Now uniqueness of solutions follows as in Walther[11]because 3—: < 0 i.e. Theorem 9.Choosing a and a as in lemmas 5, 6, and 7 there is at most one solution of equations (3.2), (3.3) and (3.4). Proof:Proposition 8 implies that 17 is a function of a. :i:(t) is uniquely defined as an integral curve of a Lipshitz ordinary differential equation. Hence :1:(a, 0(a), t) is the unique solution to equations (3.2), (3.3), and (3.4). I Corollary 10.The unique solution in Theorem 9 is a periodic solution 2:" of the delay differential equation (1.1). 13 Prooszhe proof in [11] applies with small changes related to our choice of initial conditions. Define 1(1) 2 ya(1 — t), and y(1) = :ra(1 — t). Then it is easy to verify that 93(1) and y(t) satisfy the same differential equation as xa(t) and y°(l) do and :1:(0) = y0‘(1) 2 17(0), y(0) 2 20(1) : 0, where we used the Hamiltonian structure of equation (3.2). Uniqueness of solutions to Lipshitz ordinary differential equations now shows y°‘(1—l) = 3°(t). A similar argument shows ya(—t) = —y°‘(t). Hence 21:01:31100—1): 3.21004) = anew—1))- “’0 define the first positive zero of:1:,2¥5 by 33(23) 2' 0 and :r(t) > 0 for all t E (0, 23). Similarly 23, the second zero of :1: is defined by 3(23) 2 0 and :1:(t) < 0 for all t E (2‘13, 235) . We next Show how to choose (15,61, and (7 to garauntee existence of these zeroes. Proposition 11.For any interval [a0,a,v] with do > 3 there is an 6 so that if [96 — :L‘alco < 6, then 23,23 exist and |2[IS — 23] > 1. \\"e also have ir'(2[b) 75 0, :1:’(2.3) # 0. Prooszhe flow generated by equation (2.1) is uniformly continuous in the CO norm topology with respect to a, <35, and t in bounded time. We have 30(0) > 1/2, 1:0(2) < —1/2, :1:°’(~l) > 1/2. These two facts imply the existence of 60‘such that if [¢—$a[Co < 60 and :1: has initial condition gt, then :r(0) > 1/2, 22(2) < —1/2, 12(4) > 1/2, consequently by the intermediate value theorem 2? exists and there is a t2 > 23 such that 2(11) 2 O. The existence of 23 will be shown later; we have to rule out the possibility :1:(t) = O for all 1423,11). Since [:c’(t)| < 2aN for all t, 23 > l/aN, consequently the 1* defined next is nonzero. Define 1* = inf|¢_rolco<€0 2‘115 and choose 6 so that |¢S — malco < 6 implies 96(1) > 0 if! E [—1+t"‘,t*] (if 1* >1 we are done). Now :r’(2[fi) 74 0 by the definition of 23 and Of the flow. Therefore 2; g 11 exists. The properties of 23 now follow directly from the fact that xf(:1:) g 0 and that the solutions are slowly oscillating. I 14 We define C*[—1,0] to be {¢EC[—1,O] : ¢(——1) = 0} and P : B Q C*[—1,0] ——> C'*[—1,0] by P¢(t) 2 13(23 +1). Ilere B = {95 : |¢—xa|co < 6}, :r is a. solution of equation (2.1) with initial condition qu and the 6 is chosen small enough so that 23‘ is well defined. Walther[ll] p.273 shows that this map is compact and differentiable in a neighbourhood of 3". Since P is a continuous compact map from a neighbourhood of 1'0 to C”, it has a. local Leray-Shauder degree. See, for example Nirenberg[12] Chapter 111. Definition 12.The index of :r" is defined to be deg(:1:°‘, (id — P)]B, 0). To calculate this index we need to define some more maps. \V'e define L' : C'*[—1,0] —> C*[—l, 0] by: U(1D)(t) : 113(4 +1), where ”113 satisfies u'(t) = afa'(:1:a(t—1))u(t —1) (3.17) and u|[-—1,0] = t9. U determines the derivative of P at 1:“. Let /\ be any complex number. il[(/\) is defined to be the algebraic multiplicity of )1 as an eigenvalue of L'. \Ve quote \V'alther[11] p. 271. Theorem 13. The index of 3:" is (—1)Z|A|>1M(A) if 31(1) 2 1. \V'alther shows for a class of functions f, that the index changes at least once as a —> 00. \\"e Show for our f0 that the index changes at a large finite number of large 01’s. Our work can be interpreted assaying something about the placement of the set of delay differential equations with bifurcating periodic orbits. Roughly our work shows that this set intersects a. straight line in the space of delay differential equations in many places. 15 4. Linearization Around the Periodic Solution Since P is compact and differentiable, its derivative is compact and has spectrum consisting entirely of eigenvalues. Following \Valtherfll] we search for eigenvalues equal to 1 by re-writing equation (3.17) as an ordinary differential equation with boundary conditions giving eigenvectors for a square root of U. Let ll’t/J(t) 2 113(2 + t) for all t 6 [—1,0], where u satisfies equation (3.17). ‘Then ”'2 = U and the relation (IV — A)(1V +/\) = U — A2 shows that all eigenvalues for U arise as squares of eigenvalues of \V. Define u,\(t) = u(t) where u satisfies equation (3.17) and u(t + Q) = /\u(t). (4.1) Define 2(t) =2 u,\(t +1). \Ve will drop the subscript /\ and write 21(1) for u)‘(t) and 2(i) for 2,\(t). We next define times t when the flow defining it changes its nature. alazinf{t:t>0and xa(t)<1—a} aa 2 inf{t : t > 0 and xa(t) < 1} am =inf{t:t>0 and 3:0(t) <1+0'} blazinf{t:t>0andxa(t—1)>1—a} bazinf{t:t>0andxa(t-—1)>1} bga :inf{t:t>0and 2:0(t) >1+a} Then equation (3.17) becomes: r —1 [0 /\ ]& 1ft 6 [0,010] l 0 C(a,/\,t)fl ift E [(110, 020] __ —1 run) 2 oz< [‘1] *0 J 21 ift 6 [£120,510,] (4.2) G(C¥,A, [)1] lft 6 [610,620] _—1 [0 “l ]a ift€[bga,1] K 16 where aw, .) ._. 0 WW» Equation (4.2) can be written as a’ = aG(a,A, 021(2). (4.3) Then 2] satisfies: _ _ /\z(0) 11(1) — [11(0) J. (4.4) Since fa is almost piecewise-linear, it is useful to change equation (4.2) to a statement about naturally defined maps. Letting 11 satisfy equation (4.3) we define the following five maps from R2 to R2 : R0210 = 21(a10) where 22(0) = 210 20,110 = mag...) where ma...) = {.0 1102-10 2 12(5)...) where 21(a20) = a0 532110 = 11(b2a) where Who.) = 110 12-0120 = 12(1) where 11032..) = {10. All these maps are C1 in a if the am and bla are smooth functions of (1. Rm [[0, [La are rotations and solutions of hyperbolic linear equations, and ISO — 1|, [2”; — 1| turn out to be negligible up to order a. The index of P as a function of a can change only as A, an eigenvalue of \V, goes through -1 or 1. We follow the outline of the proof in Walther [11] that /\ does not pass through —1. Let 6(fl(t)) be defined by tan(0(fz(l))) = u(t)/z(l) and the integer part of 0/27r is the winding number of 12(t) about the origin. We will often write 911 for 0(12) in the rest of the thesis for readability. Lemma 2.) = -1 and a > (10 and 0' < 00(00) implies |6fl(a10,)— 9a(bga)| <32E 17 Proof: First note that 1] satisfies {1’ z a [ 0 1] a —1 0 fort E [1120,1710]. We have 020, — bla = 2(1/2 — am). Thus lgflfbia) — 671(02011'5 205(1/2 ‘ am)- As in lemmas (2.2), (2.3), and (2.5) |6a(b20) — 93(510113160 |617(a2a) —— 9&(a1a)| 3160. Taking a large and then a small we have l9fl(b2a) — 011(a1a)|3|611(bla)— 612(a2a)|+160‘ S 20(1/2 —— am) < Mltl since 0(1/2 — a1a)£ 160 for sufficiently large 01. I Theorem 3 .For a > g, a chosen as in lemmas (3.5), (3.6), and (3.7) there is one solution to equation (4.3) with boundary conditions (4.4) having /\ = —-1 and this solution has algebraic multiplicity 1. Proof:(i) We show that dim ker (\V 4— id) = 1. Assume, for contradiction, that dim ker (W + id) ;£ 1. Then dim ker(\V + id) = 2 since :13" 6 ker(W + id). If dim ker (W + id) = 2 then for any initial condition 11(0) we find that In particular 11(0) 2 [H implies 11(1) : [_11 J. However, this implies 12(bga) = [“65] for some 5 > 0. Similarly we have 11(ala) = [g] for some fl > 0. Therefore we have 37r Imam) — 022(1)...» = 7 18 However Lemma. (4.1) says l0fi(ala) _ 611(b20)| S g: a contradiction . (ii) VValther[11] ’s proof applies without change to show that the algebraic multiplicity of (\V + id) is 1. I 19 5. Satisfaction of a Necessary Condition Since we have seen that A does not pass through -1, we expect the only way that. the degree of P could change is by A going through 1. (If A crosses through the unit circle at a point where ReA 76 0, A also does, and the pair does not change the degree of 1:" ). In this and the next section we show that A must often equal 1 in the boundary value problems (4.3) and (4.4). Thisfact, while promising, does not in itself force bifurcation of .12". We will only be able to show bifurcation in Chapter 2. We now consider A = 1. The boundary conditions (4.4) split into two conditions; one which can be satisfied at any large a by a good choice of 17(0), and one, studied in the next section, which restricts the possible a. A sketch of a typical solution to equation (4.3), Chapter 1 with A = 1 is useful for motivating the remaining constructions in this section: ”2. U=Z t; Figure 3 20 The boundary conditions (4.4) appear as the requirement that 11(0) is the re- flection through the line u = 2 of 11(1). Figure 3 suggests that a necessary condition for the boundary condition to be satisfied is : I 1101101)! = l 170%) |- (5-1) This is true. Lemma 1.1f 11 satisfies equations (4.3) and (4.4), then 11 satisfies equation (5.1). Prooszoundary condition (1.4.4) says that Therefore Now RC, and [La preserve norms and map 11(0) to 11(a10) and 11(bga) to 11(1) respectively. Hence [1701101) l = | 3012(0) I = 11-10)) I = l 21(1) l = l [1-021(b20) l = 117(5201) |" For all sufficiently large (1 equation (5.1) can be satisfied by some choice of 11(0). To show this mainly involves explicit calculation of HO, and an appeal to the closeness of Ea and E; to the identity. The next two lemmas estimate Ea — 1 and Z; — 1: Lemma 2 .There is an do such that for any a > 00. 4a 40' lala_a2al<— lbla—b2al<— O! CY 40 40 lala_aal<_ lbla—bal<— Cl CY 0' 40’ 4 Ida—agal<_ lba—b20l<—a_' 21 Proof:\\"e estimate lam — agal. All the other pairs may be estimated in the same way. If! E [alml], then ya(l) > ya(a1a). f0 is increasing on [ya(ala), ya(aga)]. Thus ift E (am, 020,]. Therefore 160(1) - 13001101) > affl/"(alaDU — 01a)- Setting 1 = aga, 3:0(0201) — $001101) :: 20 > af(ya(ala))(02a — 01a)- Ilence 20' af(ya(ala)) Choosing (10 so that f(yao(aa0)) > %, we have la2a _ 0101' .< 40 la201 _ alal < 01' if a > 010, since f(y°‘) is an increasing function of a. I Lemma 3 .There is an do such that for any 01 > do and a < 00 |Za—1|<160and|Z;—l|<160. (5.2) Pr00f1VVe will deal with Ea since 2; may be estimated in the same way. We have 20110 = 11(aga) where 11’ = aG(a,n,t)11 and IaG(a,17,t)| < 2a. Then Gronwall’s inequality implies | 1101—1101101) | S | 6200—01“) — 1 l l 110110) l- Setting 1 2 1120,, and choosing a0 as in Lemma 2 we have 40 lam—2201 s leg°(“°‘“‘°)—~1 I I aol s 40751210) 22 since Lemma 2 says Iago, — alal < 35-, and |et — 1| < 21 ift < 1/2. Rewriting the last equation we have l 20170 - {to | 3160 l 170 |- (5-3) We re-write equation (5.3) as | 20, — 1 | < 160‘ and we are done. I we next calculate Ha. Lemma 4. cosh(l) — sinh(t) Hang Z —sinh(t) cosh(t) 110 + K(a)0' where t = 2a(1/2 — am). Proof:\\"e may solve the equations defining Ha explicitly giving the expression in Lemma 4 with t = 201(1/2 — aga). Lemma 2 allows us to replace ago, by am yielding Lemma. 4. I We next show that for any large 01 there must be a choice of 11(0) satisfying equation (5.1) by using the bounds we have derived on 20,, Z; and the calculation of HO. Lemma 5.There is an 00 > 0 such that for each 01y > do there is a 00 such that a E [armory] and a < 00 imply | 11(620) I = | 11(a1a) | for some choice of 5(0). Proosz'e note that a(1/2 — ala) S 817 if a 2 cm. This fact follows from arguments similar to those used in the proof of Lemma 2. Now let t = 20(1/2—— 0101)- Then [i] = [52111) 1:21.111] [11+ W = (1—1)[i]+0(12)+ K(oz)a. 23 Therefore 23,110, 2., III 2 (1-1) III +O(12)+K(a)a since I20, — 1| and I2”; — II are of order 0. Thus if 11(ala) 2 III, then 11(1):...) 2 (1 ——t)11(a10,)+0(t2) + K(a)a. Similarly if 11(ala) = I—OII then 11(b20) = (1 + t)11(ala) + 0(3) + K(a)0’. Since t = 2a(1/2 — 1110,) g a, we may choose C10 large enough and a small enough so that t2 < t/3 and then choose a smaller, if necessary, so that the term of order a is less than 2a(1/2 — ala)/3. \Vith these choices if 11(a10) = III, then I 11(b20) I < I 11(ala) I. Similarly if 11(a10) 2 I31] then I 11(bga) I > I 11(a10) I. Therefore by continuity of EEHQEQ : 122— > R2 there is a ,5 E (0, 2) so that if 01a...) = I‘lfflI. then 1221251 = 1221a...) l-- Define 220(1) by I aa(0)| = |210(1)| = 1,110‘(0) > 0. Lemma 6.110(1) is uniquely defined under the conditions in Lemma 5. Proofiff not, by linearity of 2311020, all choices of 11(0) would satisfy (1.4). However the proof of Lemma 5 shows that 11(0) 2 ill never satisfies (1.4) when 01 > aoanda < 00,- 24 6. Determination of Bifurcation Parameters We need to define basic geometric quantities for our later lemmas. We define 00(01) = arclength {11“(3) : 01a 2 s 2 0} = arclength {11"(3) : 1 Z s 2 bga} and 61(0) 2 6110(a10). Theorem 1.For each N > 0 there is an do > 0 and a 00 > 0 such that there is a set A of least N a’s {on-H; which satisfy equation (1.3) and boundary conditions (1.4) with A = 1 for any a < 00 . Proof: The proof of Lemma 5 in the last section shows 61 — 60 that EZIIQEQ110(a1a) : 110(b20). Since IE”; — 1|, I20, — 1| 3 160, we have l1[a&a(ala) I : I fla(b1a)l+ KW)"- Hence, using properties of HO 6110(010) + 6110(1)”) = 7r + 2 m 71' + K(a)a where m is any integer. Since IE; — lI, I20, — 1| 3 160 and 6 is uniformly continuous outside a. neighbourhood of the origin, 6110(a1a) + (9110(b2a) = 7r + 2m 7r 4— K(a)0. Therefore 61—10(1) :- gl—LOUJQO) - 60 = 7r—91—90 + 2 m 71' + K(a)0. The boundary condition (1.4) yields Using 6110(0) = 91 —- 60, and our expression for 110(1) in terms of 61 and 00 we find 01—90—1—7r—91—90:g+2m7r+K(oz)a 7r—260 = g + 2m7r + K(oz)(7 60 = §+m7r+K(a)0‘. 25 Now 00 = a am, an unbounded function of 61 since £2E< 00(0) < O: for large a. To find our 01,- take 010 large enough so that there are solutions of the equafion 90(a) = i: + 2m7r for at least N+2 integers m, this can be done because % < 90(a) < 01 and 60 is continuous. Then choose W i _1 — max K(a) . 2 or e [00,01N+2 ] \Vith this choice we will have at least N solutions remaining of by continuity of am as a function of oz and the Intermediate Value Theorem. 1- Corollary 2.There are choices of 00, a, and (10 such that &a(a1a) : [0(1/21—a10)I+O((a(1/2—a10))2)+K(a)a fora E -4 anda < 170. Proofi I Hal—lafala) I : I11"‘(a10,)I "I" [{(0‘1‘7 implies . —a _ —ua(a1a) 11.21 (a...) — a: I 201a...) I+K(a)a (6.2) by properties of Ha. \Ve concentrate on the positive case. The negative case is ruled out for large 01 since then Ha is approximately the identity matrix. A calculation using the definition of Ho gives _a _ 1 —2a(1/2—a1a) _a 110,11 (1110,) _ I—2a(1/2—a10) 1 I11 = I “001211)—20(1/2_a10)2a(a2a) I (6-3) —2a(1/2 — ala)ua(a2a) + 2001212) 26 neglecting terms of order 0((a(1/2 — 1110))2) and K(a)a. Setting the first compo— nents of equations (6.2) and (6.3) equal ma...) 2 0(1/2 — ala)z°‘(ala) + O((a(1/2 — 1110))2) + K(a)a. (6.4) Using I 11a(a1a) I2 = 1 we have ”(0100+ 0000/? - 010111214" K0110) = 1- Therefore 201a...) = 1+O((a(1/2—a10))2)+K(a)a. Hence, using equation (6.4) 110(1110) 2: a(1/2 -— am) + O((oz(1/2 — (110))2) + K(oz)cr. Using IE; — II, I20, — II S 160 we find equation (6.1). I We will write Rot: where :1: is any real number for the matrix cos.1: —sin:1: sin .1: cosa: ' Corollary 3.For choices of a and 0' as in Corollary (1) 1 110(0) 2 922- III +O(a(l/2-—ala)) +K(a)0'. (6.5) ProoszVe have 01am 2 71/4 mod 2m 7r for 01 E A. Therefore 110(0) = R_afia(ata) = (Rat¥+K(a)a) lam/21— (110)] _ 1 _ %I1I+ 0(a(1/2 — am» + K(a)a- - Chapter 2 CHANGE OF STABILITY FOR x’(t) =01f(:r(t — 1)) 1. Introduction. In this chapter we show that bifurcation occurs at all the 01,- constructed in chapter 1. We do this without perturbing generically as in Mallet —Paret [6]. The techniques he uses do not allow us to perturb remaining in the class of equations of pure delay. \Ve would need to consider equations of the form :1:'(t) = f(:c(t), :1:(t — I), :r(t — l),:1:(15— 3), . . . :1:(t — 1)) 11 TI which leaves unanswered questions about the effect of pure delay and symmetry on this family’s passage through the set of bifurcating dynamical systems. Our explicit calculations show the existence of bifurcating periodic solutions of the unperturbed family, and the calculations are algebraic in character, showing that questions of stability can be determined symbolically - an alternative and check for numerical methods for large a . 27 28 2. Definition and Discussion of a Bifurcation Function In Chapter 1 equation (4.3) and boundary conditions (4.4) gave a way of finding the spectrum of the linearized flow around the periodic orbit of equation (1.1.1). One way of studying bifurcation near 01,- is to choose a function of a and A which vanishes when A = 1 , is differentiable and related to the flow generated by (1.1.1). Walther [11] shows that a function qa(A) defined in terms of the fundamental solutions to equations (4.3) and using equation (4.4) will give us bifurcation results; we introduce functions q,-(a,A) identical to \Valther’s up to multiplication by a 1 constant. Our choice of q,- is more closely tied to the details known about the bifurcation points (2;. This specialised choice of bifurcation function is effective only because of the extra information we have in this case. For each i an integer greater than zero define __ ‘ 11(1) - Az(0) 11*(1) -— Az“(0) “0“") “ d“ l 2(1) — um) 2*(11— 2110) l where 11(t) and 11*(1) satisfy equation (1.4.3) and 01' {1(0): I”Q,I and a‘(0)=I(1)I 2 Our q,-(A) = nga(A) where C,- # 0 and qa(A) is defined by \ValtherIIl]. \Ve - quote the relevant corollary in WaltherIIII as a proposition with slight changes for our situation. Proposition 1.Suppose q,-(a,1) # land the algebraic multiplicity ofker (\\'+id) = 1. Then the index of the fixed point :5" of P is given by 1Hd€X 2:0 :: Ci(_1)ZA<_lj(A)(_1)Zl 0 such that for all 6 < 60 index 10““ # index 10““ if q,- satisfies: (1:101:11 = 0 (i) (993' a—a( 111) 7£ 0 (22) ('3 0—:(a., 1) 4 0 (m) Proof:(i), (ii) and (iii) imply there is a simple zero A(a) of q,- in a neighbour- hood ofoz = 01,-, A = 1 . The Implicit Function Theorem, (ii) and (iii) imply 3—: 35 0. We handle the case where 3—: > 0 , a parallel argument works for 3% < 0. Since 2% > 0, A(a,- — e) < 1 and A(a,- + E) > 1 for 0 < e < 60. \\"e know from Theorem 3 Chapter 1 that j(-1) = 1 always, consequently the hypotheses of Proposition 1 are satisfied if A(a) 75 1. Since A(a,- +6) 7f 1 we can use Proposition 1 to calculate the degree of 1:“ for a = 011 i e . Rouche’s Theorem implies the degree has constant contributions from any roots not equal to one at 01,-. It is now clear that index 10“” # index 10'“. I We therefore must calculate 831(1)“; 1) and a3‘1A—'(()z,-,1). I a 30 3. Technical preliminaries Let N15:(0) = 1(aa), N2513(0) = :E(a20,) where 5:(a10,) 2 33(0), 1(aa) : 21(0) respectively and In the next two lemmas we show that these maps and their derivatives with respect to a are small. This will enable us to calculate the derivative of am up to order 0'. \Ve need this derivative to find the quantities %‘1 and 6710-. To bound £01m — a0) and %(aa — ago) we need the following lemma. Lemma 1.For all or > a0 and a < 0‘0 I (am. — aa)I S K(oz)a (3.1) 3 5a 31 1< K1 ,, (9a a0, —a20) _ a a 8230 62:" 6010 x | (,0 (11+ 8, (MR—I s 1(a). for i. E [(110, 02a]. Proof:\Ve prove only the first case; the second is proved similarly. The third inequality is proved using similar arguments to those for the first two. We have xa(ala) 2 1+ 0 and 10(00) : 1. The Implicit Function Theorem says that 6am 6m“ 6350 -1_ 8—30: -0‘ a _1 (,0 = a—awintwtami) — (,0 (axaxafly < 1...)» where 6—3? exists since the numerator is non-zero. A similar equality holds for @19- 80 yielding a a ‘a—(a —a 1: TLQWIO) “ iii—(a0) 8a 1" 0' Olf(ya(ala)) af(y°’(aa)) 4 6 6 3 31111065115?“ — 55101251 4 6 + ———x°‘(aa)lf(y"(aia)) - f(y°‘(aa))l- aaa 31 For the last inequality we let A : 52—30101”), B = af(y°‘(a1a)), C = 8;: (a0), D = af(ya(aa)) and used the algebraic equality .4/B — C/D : (:ID — BC)(BD)—1. \Vith our choice of A,B, C and D we have B, D> 1/2, so we can use the identity AD - BC 2 A(D - B) + B(A - C) to justify the final step in the inequality. Using the Mean Value Theorem and the fact that f(y°‘) has derivatives bounded by a we find ('3 1 a 0:13“ 18:13" Iégmla _ “(1)13 361—6; (X) I010 " aal + 33—01 (aa1Iallala _ aal (3-2) , r , 81:0 where x E [almaaI “e now bound 5; and a 0230 _ 6 , _ 6:2: an I 6—Cl- (t) — 80$(17 17701) _ 617(1) 1770060 +1.7: (117,70!) (33) Therefore 6 a”: t) = @3- 592 + oz2 if(y(1,n,at)) l- (3.1) $5; an 601 a: Lemma(1.2.2) yields 3—: < 62"" and 3—2:, < 2062‘It as part of its proof, and Lemma (1.2.5) yields 677 —1 IEEI < K(a0)a(cosh(aala)) . Equation (3.3) therefore implies I‘g‘i (ala)I S 20'6””K(010)01(cosh(aala,))-1 + a2. (3.5) 0 Equation (3.4) implies (923—1. (t)I S 202K(c10)(cosh(aa10,))—1 + a2 t a (3.6) —1 S 2021((ao)(cosh(%)) + (12. Substituting these bounds in equation (3.2) yields (9 —1 —(a10, — aa)I S 2aK(aO)(cosh 3) [1+ 60am] I010, — ac, I80 3 . (3.7) S K(a)Ia10 — aaI S K(a)a.I 32 Lemma 2. 8N.- . I-a— < K(a)a IN,- - II < K(a)0 IDN, — II S K(a)a. or ProoleDNi — II S K(a)a and IN,- — II < K(a)a follows from appropriate variational equations and Gronwall’s inequality as in Chapter I. We estimate 9533-. The estimation of 9355/52 is similar. Define 53*(t) : 11(a0Z +t). We see that 513* satisfies the same ordinary differential equation as as :1: does and differentiability with respect to t and (1 gives 0M_34 _ _a* _ _- __1 _ (9C! — 60:1: (ala a0) -— (9a (ala aa) + x (010 aa)aa(ala (Ia) where 65‘“ 0 f’(xa(t))I -1. _ 2 I a 0 + F’ a... [My (1)) 0 I IFI < 2, and By Cronwall’s Inequality 61* 5; (am, — aa) < 620(“1°—a°) - 1. Therefore 0V 1 1 S4(a10 — a0) + K(a)a (901 4 S 4—0; + K(a)a S K(a)0 a where we used Lemma (2.1.2 ) to bound (01a — aa). I flfx) If p ; Rn ._. 11", then DF is defined to be (33) if F = ; fut”) Proposition 3.1f .4(a) and B(a) are C'1 functions of a mapping R" to R” then a (98 3.4 -a—a(.-1(a)(B(a))) — 0411(B(O))(a—a(01)) + (531(30'11- Prooszhis follows from the chain rule. I 33 Lemma 4. aala 6a a I = Qalaa(1/2 — 01a) + 0(5 “ 01a1+ K01)” Proof:\Ve have 5:”(aga) and 5300110,) related by 1'\’21\’1:1':"(a1a) : 10(020). Us- ing Proposition 3 and Lemma 2 repeatedly we find i:Z'(a20,) = -a—:E(ala) + K(oz)0’ = -0—:E(aa) + K(a)0' aa 80 6a Letting Q = 5(aa), fl = distance(Q,line through (0,2) and (2,0)) and 6 2 angle between OQ and the line y = x we can draw Figure 4. ‘32 Figure 4 From Figure 4 it follows that 13 I 9—6 19 = arctan 34 Therefore ___ Hfi_ 2 sec (9 )86: 7—”) 610+ «5+ 0(3 )) (3-8) We relate 6 and B by 6 f¥fl 9 a = tan 6 2/ sec2 sds S / sec? 6 S 6 sec2 6 o 0 since see is increasing on [0,1]. Multiplying by cosg(6) we find that 6 fl 2 . ,6 —< cos 6 =sm6cos6<6 5 2 6 = — + 0 ,3 . \/§ (. 1 Since IN,- — II S K(a)a and I%I S K(a)0‘, we may replace Q = i(aa) by :E(a20,) and i(a1a) respectively in determining the relationships between 13, 6, a, and all derivatives. (Here is where we use Lemmas I and 2.) The results of explicit calculations are Removing 6 we find amp—mg: amw+Kmp+0m5 on) MIC: 66(010/2 -— a...» = —1—(1 + £)il¢§ne‘a“l°] + K(a)a + 0W). (3.12) 6 This gives us relations between 17, 5%, and 01. Expanding and simplifying equation (3.12) we find that .1. 601a __1. ‘0910 _ 2. fl 2 — am — a 661 — 26 I naa(aa1a) + ac¥I(I +0(fl)) (3.13) We also have ncosh(aa1a) = I — 0. Therefore 271 cosh(aala) + na—a(aala) sinh(aala) = 0 (3a which implies (917 6 (9a na—a(a‘ala)tanh(aala). (3.14) Equations (3.13) and (3.14) together imply 1 60101 _ —aa1°_7l_a_ 2 _ (11a _ 01 8a __ —e 2 aa(aala)(1+tanh(aaia)(1+ 0(3)) which implies 601a __ Q —aala 2 aa “ 2“”26 +0”) I - 2201aa(1/2—ala)+0(fl2)+0(§ ‘ala) (310) aaala = 2a1aa(I/2 — 01a1+ 0(1— 01a)+ K(a)cr 8a 2 since 6 S 0(1/2 — am) 4— K(a)a. I Note 5. a2(I/2 — am) S (re—“1° + K(oz)a by equation (11) and hence is less than K(a)0' for large 01 if 0' is fixed. 36 4.Ca1culation of Properties of Bifurcation Functions. —1 \Ve define = A—1 and .4 = 0 fl . \Ve abuse notation writin q,- a,;1 for ’1 IL 1 0 g Qi(C¥,AT1). We define 111(01, )u) = R—a,pZ;,uI[a,#Ea,uRa,p- We rewrite qi(a,p) in terms of the maps 12-0,,“ 2* Ha,,,,za,,,,Ra,,.,.—1,, as fol- 01,111 lows, dropping dependence on ,u where possible: 210(1) = 11_a2;11020110210(0) = 111(a,,u)11°‘(0). Therefore 9161,11) = det ((3161.11) — "lu)X1:((i1[(011/1) - 4162(2). Then 3-1-11—dttaolt-11-41 (111w 1)--1>) aaqt at) — 6 aa 1 an ‘ 1X17 * H ‘ 11 X2 13 + det((M(a,-, 1) — .4,,)X1, a—a—(AI(a,-, 1) — .10).?) Using the definition of X1 and a,- this implies (cm-,1) = det(—(Z-(11[(a,-,I)— -—11)X1,(fi[(a,~, I) — .41)X2). (4.1) 63—02% (90 Similar calculations yield 39401.1”) 2‘ dct ("Q—(MW 11)- :1 1X1, (111011: 1) — 4113(2). (4?) a” all 7 [1 / at ,u = I. Explicit calculation of (11!(a,-, 1) — .~11)X2 yields (M(a,~,1) — A‘111X2 \/2I_11I + O(a(I/2 — 01a))+ K(a)0'. (4.3) 37 We replace the determinants by a dot product with an appropriate vector as this simplifies some calculations. It may be easily verified that equations (4.1) (4.2) and (4.3) imply (73?;91(0h1)(1+ 0(a(1/2 — 1110.)) + K(a)a) = 5%(M(aa/11X1)- III (44) a I .. $9i(01,1)(1+ O(a(I/2 — a1a))+ [{(a)0) = 111(a,,u)X1- .4“). [1] (4,0) atozza', and/1:1. Lemma 1.At a : oz,- ——(11_(9 2*]! 2 RQX1)=1W—a[(iz;)11 11 +H (— a —z: )11... “10,311., 8a Ba Ba +(—a—aHa)RalX1+(B%R—Q)HaRaXI+K(a)U' 8a Prooszhis is a calculation using Lemma (3.2) repeatedly and the facts I20, — II, IEZ—II S K(a)a, D20, 2 So, DZ; 2 22,0110, 2 Ha, D120, :2 Ba, and DILO, : 11-0, (since $0,532,110, and 1L0 are linear). l Proposition 2. For any 110 E R2 _0_ _ __Ialoz—‘a2aI 0 (90120110 _ U Iffyfaia11U0201aCYf1/2“010)I +K(a)a 12* {10 : _Iala "‘ aQQI f(:r(bla))202alaa(I/2 — (11a) + I((O)0' 8a a a 0 Proof: To 'afind 39302 and,9 —%2* up to order a notice that fort E [0101,0201], by the definition of 5.3a we must look at -, _ 0 Watt» - _ - 1. ”hm/0(1)) o 11““ la‘1(z"(t)—1) 0 I“ Hence, replacing t by t - am fi’_ 0 —-1I_ 801 u 38 with Further dividing t by a we find @’_ 0 —1 _ 0 0 _ aa ‘0 U—1(ma(t)—I) 0 ”+0“ 6;;(ala+at) 0 u 0 —1 _a__2_2_ +0” 0’1(:1:C'(l)—I) 0 5a fort E [0,0'_1Ia10 —— agaII Q [0,4]. Since Ia_1(1:°‘(t) —— I)I S I, the last equation becomes 6—11I __ 0 0 6a — 83x:(a101 + 001‘ The same arguments that bounded I20, -— II yield I11 — uQI S K(a)0. This 0 I 11a.(a1.,.)+0() estimate with Lemma (3.1) now implies 811 0 — i Z a . 0a() QI%(a1a1UOI+ 0(a) Noting by Lemma 3.1 that a—arQ:(alo,) = —af(y (010115630112 and substituting [2 w“;al° we find Iala — 0 a2aI U If(y(ala))u02a1aa(I/2—a10,) 6. 550120 _ ala) : _ + K(01)0', where we used continous dependence of solutions to ordinary differential equations on their right hand sides to justify the substitutions. Since 6 6 6‘ , 6 5320110“ — 6—au “(a-2a aia) = 5—:(0201 - am) + 1101212 — 01a16—afa2a ‘ “'10): we have a _Iala — QQQI 0 aagafi O _ a If(y(a1a))u02ataa(1/‘2 _ ala)I + K(a)cr. (4.7) Similarly we find £2: {10 : _Iala _ (Dal If($(b1a))2020600(1/2 — 0101)] + K(a)0'.l (4.8) 39 The calculations to determine g9; and 85% are now straightforward, we quote from the results calculated in Chapter 3 . Proposition 3. 691 a;=4¢%maUfl-6MXI+Om0fipflmfl» (1m 5q1_ _1_ _ , _ E_—\/§+O(a(I/2 ala))+1\(a)a. (1.10) Proof:See Chapter 3, sections 1 and 2.- The main theorem of this chapter says that for sufficiently small a there will be a large number of bifurcation points at large a. Moreover, these bifurcation points are all isolated. Theorem 4.For each N > 0 there is an 010 > 0, a 00 and a set .4 = {61; :2, such that a < 0'0 implies a,- are bifurcation points for equation (1.1.1) and each of the a,- is an isolated bifurcation point. Prooszirst choose 00 and 00 as in Theorem (2.4.1). These choices give us a (possibly infinite) set of at least N elements {(1,} where q(a,-, I) = 0. \V'ith these choices of a and a we have, by equations (4.9) and (4.10) fig; 1 . 2 , 5c; 2 26.065 — a) + 0((a(1/2 — at.» )+ Maw aqi 1 . , 8/1 _<_ ——\/r2.+0(oz(I/2—ala))+[\(a)a with a = a,- . We may assume 010 is large enough so that for all a > 00 O(a(I/2 — a10)) < 1% and O((a(I/2 —ala))2) S 11—6a(I/2 — am) . We decrease 00 so that we have maxaflaofl‘v) K(a)0‘ < 1%. \Vith these choices at]; I — > — — - 4. (9C! _ alaax(2 as) ( 11) and ‘6q.- 1 I I I ——>——-— —>— 1m ,1 - -1 + 16 + 16 - 8 ( ) 40 Applying inequalities (11) and (12) 6A_ 5‘11 5611—1 . aa——6a(8p) 7&0 at ,u = I. we may therefore apply Proposition (1.2) to conclude that all a,- are bifurcation points. Furthermore 33 79 0 implies that the condition q,(a,1) = 0 is uniquely satisfied by a,- for all a in a. neighbourhood of ai. Therefore the set A is composed of isolated points. A is a. compact set (it is bounded and the preimage under a continuous map of a. finite set) and hence must be finite since any compact set of isolated points is finite. I Note 4.3% is small for large a and small 0, more precisely for any 6 there are choices of a and a so that 29—: < e by arguments similar to those used in Theorem 4. Note 5.Unless one has monotonicity of some type in f, determining stability of 3:" (rather than change of stability) seems difficult to do analytically. A major problem in the analysis is to handle complex multipliers of the orbit. In the approach used here one has a boundary value problem in 4 real dimensions, which seems difficult to study. Chapter 3 CALCULATIONS 1. Computation of £3934 - a We will calculate each of the terms in Lemma (3.4.2) separately. First, the terms involving 20, and 2; Lemma 1. R [32*([{ R )+I[ £2 (11)] —01 aa a a 01 080 a 01 XI = -R Iala "a2al If($(bla))2alaa(1/2_ala) (1.1) _a (T 0 + K(o)a + O((a(I/2 — (11011)?) Proof: \Ve have, by corollary (1.6.2) 11.120111 = I‘M/21 ‘ “‘alI + 0((6(1/2 — 61.))2) + K(a)a so Proposition ( 2.4.3) implies (8a a + K(a)a + O((a(1/2 — 610))2) 22.)“... : ___... w [166.621.0612 - an» 0 (1.2) Similarly RaXI : [0(1/21_ (110)] +0((C¥(1/2 — 010))2). Proposition ( 2.4.3) implies 0 U lf(1(016))2ama(1/2_any + K(a)a + 0((a(1/2 _ (110))2) (9 I010 _a2aI ”40—626mm = 110, 41 42 or 11458525132“? 1100((60/2 — aia))2) + K(a)0 + 0((a(1/2 - 0104)?) (13) = K(a)a + O((a(1/2 — 6(a))2). Equations (1.2) and (1.3) together give equation (1.1) I Lemma 2. a a 4 R_a(1165;Ra+(a—QHQ)RQ)XI = R-a(alaa(1/2 — a”) [Di 6 —I + a—a‘(aala) [300/2 _ ala)I) (1.4) I + O((a(I/2 _ 6(a))2) + 0(5 — a...) + K(a)a Proof: First we calculate 810110,. [I = I cosh(2a(I/2—ala)) —sinh(2a(I/2—ala))I 0' — $1nh(2a-(1/2 — 0101)) cosh(2a(I/2 — 0101)) therefore 0 _ I 861101 Sinh(20(1/2—ala)) —C05h(2a(1/2—ala)) 6311a — 2G — am _ a—(‘i—a—J I“ cosh(20(1/2 - (1161)) sinh(201(1/2 — (110)) I _ 26(1/2 —(11a) *1 __4ama(1/2—aia) I _1 20(1/2—ala11I + O((a(I/2 — 6(a))2) + 0% — 6...). Using our knowledge of a(I/2 — am) we therefore have —a—Ila = 2alaa(I/2 — a10)I: 0a (1) (I)I+O((a(1/2—ala))2)+0(%—ala). (1.5) Substituting for Raxl we have (503111))30M : alaa(I/2 — am) IéI + 0((a(I/2 — a10))2) + O(% — am) (1.6) Noting that RC, is the fundamental solution for a set of differential equations and ' _ 7r . 8 . usmg aala — 3- mod 2 mrr we calculate 53-110,)“ . (9 6 —1 BERG)“ : 19—11(0010)la(1/2 —- ala)I + 0((00/2 - “10112) + K(a)a 43 so H.311“. = 3 1 6a aa (0’01“) I3a(1/;_ ala)I + 0600/2 — a1a))2) + K(a)a. (1.7) Equations (1.6) and (1.7) together give equation (1.4). I Lemma 3 . a ‘9 1 ' (a—aR—a)HaRaX1 — 53(0019)R_a [0(1/2 — (1101)] Proof: We omit the calculation, which is similar to that in Lemmai2. Using Lemmas I, 2 and 3 we find, after simplification, and using the formula (2.4.4) 5%(R_OE;IIOEOROXI) = V Iala-G2OI . = 0(1/2-01a)alaR—a [2f($(bla))4 a +4] (1.8) “'e note that Ra is norm preserving for any a so 2%: = v. III (1+ 0(a11/2 — (11011)) = 12.11'.R.III(1+ 0(“(1/2 ‘ “1“)” \\'e substitute for Ba III finding 1101111., III = 61(1/2—ala)a,a\/2‘ IgffxfbmnIII—”511°J + “‘I . I‘II+0((6(1/2 — 6.0))2) Therefore 23 =4\/§araa(1/2—ala)+O((a(1/2‘ala))2) (1'9) ' 44 2. Calculation of 6331i For the calculation of 5959: we may ignore 2a,“, 20,,“ since £20,“, 36:72am S K(a)a by Gronwall’s inequality. We therefore calculate 6 6 6 _(R-uHuRu _ Au)X1 :(ng-111HHR11X1 + R-AEI'HHRIIXI 6 . 6/1 I "fl ‘1 a [.1 X1 8 ‘ #XI at [1 = I. Explicit calculation gives us 12,, = I cos(o\/fiala) 71—Hsin(a\/;1ala)I . —\/fism(a\/flala) cos(a\/fiaia) .therefore 6 _ aala —sin(aala) cos(aa1a) . 0 I aRquzi _ 2 I—cos(aala) —sin(aa1a) —sm(aa1a) I 0 __ aala . 0 I _ 2 Rot_§ —sm(aala) I1 0] + K(a)0'. Similarly §;R_#l#:1= _OGIOR01% +sin(aala) [(1) (I)I + K(a)0j therefore 6 a C“1111 all-1111111211)“ + R-#H# $12100 : 2 (_ROHQROX1—i H-0I10R—0X1) . 0 1 0 I +sm(ara10,)(I1 OI (IIOROX1)- 11-0110, Il OI X1)- Since 1 0 110.: I0 1I+0(a(I/2—a10)) we have 6 6 —R_ [I R- H —R 511 11 #R#X1+ .U flap IJXl : —aala(Rot.}Rot§ + Rot—TxRot—Tgbm . 0 I 0 I +sm(aala)(I1 0IROX1_R_OI1 0IX1) + O(a(l/2 — a10)) + K(a)0' 45 Substituting X1 = Ida/21— ala)I + O((a(I/2 — ala))2) + K(oz)a we find 6 6 'a—fiR-uHuRqu + R-flllfl @1211)“ = 0 + 0(012(1/2 — 0101)) + K(a)a By Gronwall’s inequality and the variational equation for Ho, £110“, is of order 01(1/2 — am), therefore (9 a Elfl:I(RuI[#Ru — 1111))(1 = 5;:111ILLZIX1 ‘l’ 0(QQ(1/2 — 010)) + K(a)(f (2.3) Therefore, using (3.4.5) %— -—l— _1 a2 —a do 1 a —a (9,, -(\/§I0 I +0( (1/2 16))+K( ) ).I1I(1+0( (1/2 1,))) 6q.-___1_ (12 —a ’00-. ' ' E;— fi+0( (1/2 16))+1\()- (2-4) 3. Machine calculations The following manipulations were done on a Sun 3/50 using macsyma, a symbol manipulation program. Pieces of the calculation as well as the final results were checked against hand calculations. There is noticeable divergence in the type of symbolic manipulations done by hand and by machine, the main cause being geo- metric guidance in the algebra done by hand. /* for partial with respect to a >“/ /*.1: = a,a1= a1(a), 22 = a(l/2 — 1110,) */ al(x);zz(x); /* tell the system a1 and 22 are functions */ let(diff(zz(x),x),-2"‘al(x)"zz(x));let(diff(al(x),x),2 * aI(x) "" zz(x)/x); /"‘ tell the system the derivatives of al and 22 */ let(cosh(2”‘zz(x)),1); let(sinh(2*zz(x)),2"zz(x)); 46 let(cos(x"al(x)),1/sqrt(2)); let(sin(x"al(x)),1/Sq1‘t(2)); /* These are true statements at 01,- up to higher order in a(I/2 — c110,) */ hI: matrix([cosh(2* zz(x)),-sinh(2*zz(x))], [-sinh(2"‘zz(x)),cosh(2*zz(x))I); h:letsimp(hl); dh:1etsimp(difi'(h1,x)); dsigstar:matrix([0, yy*a1(x)*zz(x)l. l0: 0 I); rizmatrix([cos(x*a1(x)),—sin(x*a1(X))l. [sin(x*al(x)), cos(x*a1(x))I); dr:letsimp(diff(l‘1,x))§ r:letsimp(r1); rminusl:matrix([cos(x"a1(x)),sin(x"al(x))I, 1—sin).cos(x*a1(x111); drminus:letsimp(diff(rminusl,x)); rminus:letsimp(rminusl); oneone:matrix([1,l]); chil : ma1r1x([(1+zz(x))*1/sqrt(2)]. 1(1-zz>*1/sqrt<2>1); ans:oneone.(rminus. (dsigstar.h.r + h.dr + dhl ) +drminus.h.r ).chi1 factor(letsimp(ans)); /* the result: checks with hand calculation m/ 8 al(x) 22(x) (22(x) + 1) (d24) 47 sqrt(2) /x for partial with respect to p we define functions of p’k/ /"< 22 = 0(1/2 — 01a)*/ /* x = a */ /* mu = 11 */ hI: matrix([cosh(2"< zz),-sinh(2"‘zz)I, [-sinh(2*zz),cosh(2*zz)]); r1:matrix([cos(x*a1*sqrt(mu)),—sqrt(mu)*Sin(an1*Sqrt(mu))la (1 /sqrt(mU)xsin(-‘ixalxSqthmU111COS(X*31*Sqrtfmum); rminusl:matrix(Icos(x*al*sqrt(mu)),sqrt(mu)*sin(x*a1"sqrt(mu))I, [-1/sqrt(mu)*sin(x"a1*sqrt(mu)),cos(x"a1*sqrt(mu))I); /" we give ourselves easily checked original versions of each of our matrices x/ let(sin(al"sqrt(mU)xX),Sin(alXX»; let(cos(al”sqrt(mu)"x),cos(a1XX»; let(cosh(2’< 22), I); let(sinh(2 zz) 2722); let(cos(x”‘a1),1/sqrt(2)); let(sin(x* a1), I/sqrt(2 )); let(sqrt(mU)1;) let(l/mu, 1); let(l/power(mu,(3/2)))11)§ let(I/sqrt(mu), I); /" tell the system some things that are true up to order a and 22 2 26/ drmu:letsimp(diff(rl,mu)); drminusmu:letsimp(diff(rminusl,mu)); HIE—___ 48 h:letsimp(hl); r:letsimp(r1); rminus:letsimp(rminusl); /"< tell the system to simplify using the let rules we have given it */ oneone:matrix([1,1]); chiI : matrix([(I+zz)*l/sqrt(2)], [(l-ZZ)"1/sqrt(?)l); ans:oneone.(drminusmu.h.r+rminus.h.drmu ).chiI ; factor(letsimp(ans)); /x factor tells the system to work harder at simplification; without this the system produces about 2 pages of output */ substituting values for x and zz(x) as defined above we get: 222(22-2a1 x) (d35) - sqrt(2) consistent with 0 + O (alphaxalphax(I/2-al(alpha)). LIST OF REFERENCES [II S.-N. Chow, O. Diekmann and J. Mallet-Paret, Stability, multiplicity, and global continuation of symmetric periodic solutions of a nonlinear Volterra integral equation. Japan Journal of Applied Mathematics, 2 (1985), 433-469. [2] S.-N. Chow and H. O. Walther, Characteristic multipliers andstability of symmetric periodic solutions of x’(t) = g(x(t-I)). In preparation (1986). . [3] J. K. Hale, Theory of Functional Differential Equations . Springer 1977. [4] J.L. Kaplan and J.A. Yorke, Ordinary differential equations which yield peri- odic solutions of differential delay equations. J.Math.Anal.Appl., 48 (1974),3I7- 324. [5] Mackey and Glass, Pathological conditions resulting from instabilities in physiological control systems. N.X".Acad.Sci, 316 (1979),2I4-235. [6] J. Mallet-Paret, Generic propeties of retarded differential equations. Bull. Amer.Math.Soc., 81 (1975), 750-751. [7] J. Mallet-Paret, Generic properties of retarded differential equations. J.D.E., 25 (1977) 163-183. [8] J. Mallet-Paret, Morse decompositions and global continuation for singularly perturbed delay equations. Systems of Nonlinear Partial Differential Equa- tions, Oxford 1982, 351-360. [9] L.Nirenberg, Topics in Nonlinear Functional Analysis. New York Courant Institute of Mathematics and Science 1973-1974. [10] R.D.Nussbaum, Uniqueness and non-uniqueness for periodic solutions of x’(t) = -g (x(t-I)). 113.13., 34 (1919) 25-51. [III H.O.VValther, Bifurcation from periodic solutions in functional differential equations. Math.Z., 182 (1983), 269-289. [12] H.O.VValther, Homoclinic solution and chaos in :1:’(i) 2: f(:1:(t — 1)). Nonlinear Anal. 5 (1981) no. 7,775-778 49 CHIGRN STATE UNIV. LIB lllllll Ill Hill/Ill 8|! llllsl lllgllllllllll