WEQM Date LIBRA R y Michigan State University This is to certify that the thesis entitled Integral Averaging for Nonautonomous Equations presented by Robert George White has been accepted towards fulfillment of the requirements for j Ph . D . degree in Mathematics éL./i\l._ Major professor June 22, 1979 _ ".'“ VS'k 34‘)! 32'; ‘ '4’ 2:1”. E‘?“;"‘ “17:71: (3-:*'%¢1r_;71‘7'?7 l-" re- .-- - r11 1—7 v— .. .- _ E '.~“2L".A 3n _..' . OVERDUE FINES ARE 25¢ PER DAY PER ITEM Return to book drop to remove this checkout from your record. @mar/D ’7 INTEGRAL AVERAGING FOR NONAUTONOMOUS EQUATIONS By Robert George White A DISSERTATION Submitted to , Michigan State University in partial fulfillment of the requirements fer the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1979 ABSTRACT INTEGRAL AVERAGING FOR NONAUTONOMOUS EQUATIONS By Robert George White We consider the method of averaging and its application t0' bifurcation problems involving nonautonomous equations. First, a two—dimensional nonautonomous ordinary differential equation is considered. This system, written in polar coordinates, admits a change of variables which reduces the search for periodic orbits and invariant manifolds to the study of a certain canonical form. Properties such as the existence, amplitude and stability of such structures bifurcating from an equilibrium can be determined from this canonical form of the equation. An illustration of the method is offered by investigating the well known Van der Pol equation. Higher dimensional and infinite dimensional systems can be treated in essentially the same manner by restricting the equation to the center manifold. The method of averaging is used to approximate the equation of the center manifold. A bifurcation problem in a forced Wright's equation is in- cluded in order to illustrate the application of the method to infinite dimensional systems. In memory of my parents, Robert George and Hazel Caroline. ii ACKNOWLEDGEMENTS I would like to express my gratitude to my thesis advisor, Professor Shui-Nee Chow, for his guidance in the research and text preparation. His probing questions and lively discussion with me were very helpful in this endeavor. I am indebted to my wife, Emily, for her love and support during my graduate career. I also want to thank the faculty and staff of the Mathematics Department of Michigan State University for their support. Finally, I extend my appreciation to Emily Groen-White and Noralee Burkhardt for typing the final text with great care. iii TABLE OF CONTENTS Page INTRODUCTION ....... ' ................ 1 Chapter 1 THEORY OF INTEGRAL AVERAGING ........ 3 1.1. Introduction ............. 3 1.2. The Method of Averaging ........ 7 l.3. Higher Dimensional Considerations . . . 18 2 APPLICATION TO BIFURCATION PROBLEMS ..... 25 2.1. Hopf Bifurcation for an 0.0.5. in R2. . 25 2.2. Higher Dimensional Bifurcation . . . . 38 2.3. Existence of the Manifold ....... 45 2.4. An Example .............. 55 3 FUNCTIONAL DIFFERENTIAL EQUATIONS ...... 59 3.l. The Abstract Equation ......... 59 3.2. A Perturbed Wright's Equation ..... 62 APPENDIX ......................... 7O BIBLIOGRAPHY ....................... 74 iv LIST OF FIGURES Page Figure l. Bifurcation diagrams for equation (2.l 69 INTRODUCTION In [4] Chow and Mallet-Paret showed how the classical method of averaging (see [1], [4], [8], [9], [17]) can be applied to bifur- cation problems involving ordinary differential equations (ODE's), partial differential equations (PDE's) and functional differential equations (FDE's). They restricted themselves mainly to the investi- gation of autonomous differential systems. However, many nonautono- mous systems arise naturally. For example, the bifurcation of an invariant torus from a periodic orbit (see [l3]. [l5]) and periodic forcing problems introduce nonautonomous terms into the equation. This dissertation discusses how the method of averaging can be utilized to demonstrate the existence of invariant structures when nonautonomous terms are present in the equation. In chapter 1, the method of averaging is described in general. Conditions are determined under which the method can be applied to reduce an ODE to a canonical form. In chapter 2 a Hopf bifurcation problem (see [4], [5], [6]) for a nonautonomous ODE is treated. The canonical form of this equation makes many properties (direction, amplitude and stability) of a bifurcating manifold virtually trans- parent. A method is described which reduces an n-dimensional problem to a two-dimensional one on the center manifold (see [ll]). Also, a proof of the existence and periodicity of the bifurcating manifold is given. The forced Van der Pol equation (see [8], [9]) is offered as 'l an illustration of the method. In chapter 3 FDE's are considered. Since finite dimensional space cannot be considered as the phase space for such equations (see [4]. [10]), a suitable setting for the averaging to be carried out is defined. A generic bifurcation (see [3], [4], [5], [6]) for a forced Wright's equation (see [4], [10], [18]) is shown to exist when a parameter crosses certain critical values. An appendix which outlines the basic theory of almost periodic functions completes the work. 1. THEORY OF INTEGRAL AVERAGING 1.1. Introduction. Consider the two dimensional system given by 2 (1.1) i = f(x,t) = Ax + g(x,t), x e R ,- = d/dt where A is a constant 2 x 2 matrix, g(x,t) = o(|x|2) uniformly in t as |x|1+ O and g(x,t) is almost periodic or P-periodic in t. Suppose that the linearized system (1-2) i = Ax is purely rotational, that is A has pure imaginary eigenvalues, rim, with m real and nonzero. Then by making the change of variable x-+ Rx, where R is an appropriate 2 x 2 matrix we can assume that A is in Jordan form [-0 -w A = , m f 0 [_w 0 Now all solutions of (1.2) are periodic (of period Zn/w) and are represented by circles in the phase space x = (x],x2). One may consider these solutions to lie on cylinders in (x,t)-space with axis the line x = 0. These cylinders are invariant manifolds for (1.2) since any solution of (1.2) that is on one of these cylinders at any t = t0 remains on the same cylinder for all t 6 (drab). Now if 3 |x| is small then (1.1) is a perturbation of (1.2), so one may expect that (1.1) will also have invariant manifolds which are "cylinder like". To see this more clearly scale by xi+ ex in (1.1) with |€1<<1 to get i = Ax + h(x,t,e) where h(x,t,c) = e-Ig(ex,t) = 0(3). Then switch to polar coordinates by x = rNe, N9 = (cose, sin6)’ (' denotes transposition) to obtain 9 = eG(r,B,t,e) O = w + €H(r,6,t,e) where eG(r,O,t,e) = Néh(rNe,t,e) cH(r,e,t,e) = Téh(rNe,t,e) with T6 = (-sine, cose)'. Expanding G and H in powers of 2 yields '10 ll €R-l(1‘,e,t) + €2R2(r’e,t) + 00. (1.3) - m + €w1(rsest) + €2N2(r.9.t) + "' a). I where Rj and WJ are homogeneous trigonometric polynomials in sine and case of degree j + l with coefficients depending on r and t, almost periodic or P-periodic in t. That is Rj and Wj have the form 2 an m(r,t)cos"esinme I'H-m=j +2 ’ n,m20 where an m(r,t) are almost periodic or P-periodic in t. We note here that by expanding cosne and sinme in powers of exp(ike) for |k| s j + l we see that Rj and W3 haVe the form k'O. _ - Ikl§j+2 ak(r,t)e ‘ , a_k - ak k=j (mod 2) where the ak are linear combinations of the a Further (1.3) n,m‘ may be viewed as a finite Taylor development with remainder, since we need only consider a finite number of these terms in the sequel. Now if all the Rj are independent of e and t then the periodic solutions of (1.1) are on those cylinders of radius r0 where eR (r ) + 22R (r ) + --- = O l O 2 0 ° However if R1,R2,---,Rk are independent of e and t and Rk+l’ Rk+2"" depend on e and t, then one still expects an invariant manifold near the cylinder of radius r0 where eR (r > + --. + ekR (r ) = o 1 O k 0 ° That is there is a function g(9,t,e) which is almost periodic (P-periodic) in t and Zn-periodic in a so that r = r + ek+1g(6,t,c) 0 defines an integral (invariant) manifold of (1.3), in the sense that if (r*(t),e*(t)) is any solution with r*(t0) = r0 + ek+1g(6*(t0),t0,c) then r*(t) = r + ek+Ig(e*(t),t,e) for all t 6 (~w,w). O The aim of the method of averaging is to make enough of the Rj (and Wj) in (1.3) independent of e and t by means of coordi- nate changes r + F, e + G so that the approximate amplitude of any such invariant manifold can be determined. Now if we consider the higher dimensional system X. > o X f1(X:Y:t) ‘(o O W y f2(x9Y9t) where x E R2, y E Rn-Z’ A is as before, B has no pure imaginary eigenvalues, lfi(x,y,t)| = O(|(x,y)|2) uniformly in t as |(x.Y)| + 0 for i = 1,2 and f1(x,y,t) is almost periodic or P-periodic in t. Then after a scaling xi+ ex, y1+ ey the system in the coordinates (r,e.y) has the form t = eR(r,e,y,t,c) (1.4) 9 = m + eW(r,e,y,t,e) 5'= By + 89(r.e.y.t.e)y + eh(r.e.t.e) where the r and G equations have the same form as before. For a = O (1.4) decouples and the plane y = O is an in- variant manifold on which all solutions are periodic. If 0 < Isl << 1 then there is an invariant manifold, the center mani- fold, defined by y = y*(r,6,t,e) tangent to the (r,e) plane for all t and e, Zn-periodic in e, and almost periodic or P-periodic in t so that any solution of (1.4) which is bounded for all t 6 (~m,~) lies on this manifold. On this surface (1.4) becomes a two dimensional system which can be treated as before. If r = r*(e,t) defines an invariant manifold of this two dimensional system then * * * (r,y) = (r (e,t), y (r (e,t),e,t)) defines a two dimensional invariant manifold for (1.4) with the desired periodicity properties. In 1.3 a procedure is described through which the manifold y = y*(r,e,t) can be approximated to any order of e as desired pro- vided the equation is smooth enough. 1.2. The Method of Averaging. Consider a two dimensional system in polar coordinates (r,e) given by ‘So I ‘ eR1(r,O,t) + €2R2(r,6,t) + --- (2.1) - w + eW1(r,e,t) + e2W2(r,e,t) + ... 00 l where e 6 R, w is a nonzero constant and Rj and W1 are Zn-periodic in 6, almost periodic or P-periodic in t and have the form (2.2) 2 an(r,t)e"‘:9, a_ = a Inlij n n where n and Nj are integers and an(r,t) is almost periodic or P-periodic in t. The differential equation (2.1) is assumed to be smooth enough for the following calculations to be carried out. Also R3 and Wj may depend on additional parameters which are omitted since they will play no role in the following procedure, but will become important when bifurcation problems are encountered. The goal here is to describe a change of variables r + r, 6 + 5 so that in the (F,é) coordinates R1,R2,---,Rk and W],W2,---,Wk are independent of 5 and t. Proceeding by induction, suppose that the coefficients of ej for 1 s j s k - 1 are inde- pendent of 6 and t, so that k- ‘30 ll eR1(r) + --- + e 1Rk_1(r) + ekRk(r,B,t) + --- (2.3) k'"wkqm + eka(r,6,t) + CD. I ‘ w + eW1(r) + --- + 3 Then consider a change of variables of the form r + eku(r,6,t) "II II (2.4) B + ekv(r,e,t). 0| ll Its inverse satisfies r = F ' Eku(Fsést) + 0(ek+1) (2-5) e=a-e%m3¢)+mé”r Now 3 _ . k Bu - au - au P-r+€[§Fr+—e-9+‘a_] (2.6) k 3v - 3v - 8v +5E§Fr+§6 at CDT 0 II CD and the right hand side is evaluated at (r,6,t). To evaluate at (F,§,t), (2.5) must be inserted into (2.6) and (2.3). Then (2.3) becomes k- r eR](F) + ~-- + e 1Rk_1(F-) + ekRk(F.§.t) + 0(ek+]) k-l k+1) wk_1(F) + eka(F,§,t) + 0(e CD. I 'w+wfifl+°u+e and au _ g3. - - k ‘3?- (rseat) ' 3r (roast) + 0(8 ) . . . . au au 3v 3v 3v with s1m11ar express1ons for 36 ’at.’3r" e , and 5—-. $0 (2.6) evaluated at (F,§,t) is written as F = 21(1) + .-. + ek"ak_1(a) + ekRk(F,§,t) + 0(ek+‘) é = w + eW1(F) + --. + ek“wk_1(F) + eka(r,§,t) + 0(ek*‘) where (2n) Rflfiifl=RUERU+w§§fifiJ)+%¢EmU mjm mfimm)=mfij¢1+w%wamn+%%fim¢) Now u and v must be chosen so that Rk and Wk are independent of 5 and t. Consider only (2.7a) and choose u so that Rk(F,§,t) = Rk(F) since choosing v will follow similarly. Let u have the same fOrm as Rk’ namely U(r.6.t) = Z un(r,t)e”‘9, u_n = un. |n|sNk Inserting this expression and (2.2) into (2.7a) yields au ' - _ . __g. n26 Rk(r) - InfsN (3n + anun + at )e k where an = an(r,t), un = un(r,t). So we must solve . 3”11 (2.8a) an + anun + a—t— = O for O < In] 5 Nk auo - _ (2.8b) a0 + a‘f‘ = Rk(r) 10 Let us first examine the case where all the an are P-periodic in t. The following lemma holds. Lemma 1.2.1. Consider the differential equation (2.9) a(t) + amt) + 6m = o where_ a(t) is P-periodic and w» is a constant. Then the fbllowing are equivalent. (A) (2.9) has a P-periodic solution. (8) Either wP is not an integer multiple of Zn or P st . I e a(s)ds = 0 if wP is an integer multiple of 2n. 0 t . (C) I e‘msa(s)ds is bounded. Proof. The Fredholm alternative theorem implies that (2.9) has a P-periodic solution if and only if P* Job (t)a(t)dt = o for all P-periodic solutions, b*(t), of the adjoint equation 9 = (my. Thus 0 if wP is not an integer multiple of 2n * b (t) = iwt . . . , e if wP 15 an integer multiple of Zn so (A) is equivalent to (B). (B)=:>(C). If wP=# 2nk for all integers k, let n be the t integer such that Pn s t < P(nt + 1), then t 11 n t . t vP . t . I e‘wsa(s)ds = Z I e‘msa(s)ds + I e‘wsa(s)ds O v=1 v- P Pnt n . t-Pn t . P . Lwn P t . Z1e‘w(v’1)PJ e‘wsa(s)ds + e t I e‘wsa(s)ds v= O O iwn P l-e t 1_ein t-Pn P . iwn P . . I e‘“sa(s)ds + e t I tewsa(s)ds O O which is easily seen to be bounded since 0 s t - Pn < P and up t is not an integer multiple of Zn. On the other hand if cnP = 2nk for some integer k and P . S I e‘w a(s)ds = O 0 then by what has just been done, we have t . t-Pn . I e‘wsa(s)ds = I tewsa(s)ds 0 0 which again is bounded. (C) e> (B). If wP is an integer multiple of Zn then again by what has just been done, we have t . P . t-Pn . I e‘wsa(s)ds = "ti e‘msa(s)ds + I te‘wsa(s)ds O O O which will not be bounded unless P . [ e‘wsa(s)ds = O. 0 This completes the proof of Lemma (1.2.1). 12 Thus (2.8a) has a P-periodic solution if and only if t . J e‘wsa (s)ds 0 n is bounded for all n=# 0 that appear in (2.2). To solve (2.8b) we need t - - - Joao(r,s) - Rk(r)ds to be bounded. This will be the case if and only if Rk(r) = magnan]. Since ao(r,t) = meanRk] we have R (r) = meanER J. " M k We have proved the following theorem. Theorem 1.2.1. Consider the differential equation k- 10 ll eR1(r) + ~-- + e 1Rk_1(r) + ekRk(r,6,t) + O(ck+1) k-l k+1) Wk_](r) + eka(r,e,t) + 0(e CD. I -w+€w1(r) + 000 +8 where Rk and Wk are 2n-periodic in e and P-periodic in t and have the form Rk(r,e,t) = I an(r,t)em.'e Inlst Wk(r,e,t) = I bn(r,t)en’ée InlsMk 13 where an = a_n and bn = b_n and t. t. I e‘"wsan(s)ds, J e‘"wsbn(s)ds are bounded for all n +=O that appear in these expansions of Rk and Wk respectively. Then there exist functions u(r,6,t) and v(r,6,t) which are 2n-periodic in e and P-periodic in t so that if r + eku(r,e,t) F = - _ k B - B + e v(r,6,t) then i = 8R (1) + ... + .k"a (;) + ski (F) + 0(ek*‘) 1 k-l k 5 = w + eW](F) + --- + ek-IWk_](F) + eka(F) + 0(€k+]) where Rk(r) = meanERk(r,e,t)] t,6 W (r) = meanEW (r,6,t)]. k t 6 k Now if the an(r,t) in (2.8) are almost periodic in t then again we must solve equations of the form a(t) + tout) + B(t) = o where b(t) must be chosen to be almost periodic with mEblc: mEa], (see Appendix). The variation of constants formula yields 14 . t . b(t) = e'Lwttc - I e‘msa(s)dsl O which is seen to be almost periodic in t if and only if t . (2.10) Joe‘wsa(s)ds is bounded. Note that -w cannot be a frequency of a(t), since if this were the case then meanEewta(t)] + O t which implies that the integral in (2.10) is unbounded. Thus c must be chosen so that -w is not a frequency of b(t). Taking t Lms c = meant] e a(s)ds] t 0 yields meantewtb(t)] = o. t Further we must show that b(t) possesses no frequency which is not a frequency of a(t). To this end suppose A.+--m is not a frequency of a(t). Then - T . T . t . meanEe"Atb(t)l limt§f e"(x+w)tdt +‘%J e"(x+w)tJ e‘wsa(s)dsdt] t T*” O O 0 T . t . lim-%I e"(x+w)tJ e‘wsa(s)dsdt T-t0° 0 0 T - T . lim 11 e‘wsa(s)J e'L(X+w)tdtds T O s T-roo 15 . T . . . Téfi'lim if e1"”Sa(s)[e"“(}‘+“’)T - e"(x+w)slds T*” 0 -£At ‘;£__[e-L(A+w)T A+w a(t)]l meante‘“ta(t)l - meanEe t t Thus A 2 mEb] ‘if A 2 mEa]. The following lemma has now been proved. Lemma 1.2.2. The equation a(t) + iwb(t) + b(t) = o where a(t) is almost periodic and w is a real constant, has an almost periodic solution, if and only if Iteiwsa(s)ds O is bounded, in which case b(t) can be chosen so that mEb] c: mEa]. Unlike the case where the a are P-periodic in t, a0 n having mean value K does not imply that t Joa0(s) - K ds is bounded. Thus the boundedness of this integral is required for the averaging to be carried out. We have proved the following theorem. Theorem 1.2.2. Consider the differential equation in polar coordinates k-l k+1) % eR1(r) + --. + e Rk_1(r) + ekRk(r,6,t) + 0(e k-] k+1 ) (D. II w + 8W1(r) + ... + e Wk_1(r) + eka(r,B,t) + 0(e 16 where Rk and Wk are 2n-periodic in e and almost periodic in t and have the form Rk(r,e,t) = I an(r,t)e"‘e Inlst Wk(r,6,t) = I bn(r,t)en£e InlsMk where a_ = a , b = b . Suppose that the following integrals are bounded in t. t . I e‘"wsan(r,s)ds, fOr O < In! s Nk t . I e4nwsb (r,s)ds, for O < In] 5 Mk 0 n t _ - [0(a0(r.5) - Rk(r). b0(r.s) - Wk(r))ds where Rk(r) = meanERk(r,e,t)1 6,t W (r) = meanCW (r,e,t)]. k k 9,1: Then there exist functions u(r,e,t) and v(r,e,t) which are Zn-periodic in 6, almost periodic in t with m[u1<: mERk], mEV] c mEWk] so that if r r + eku(r,e,t) e + ekv(r,e,t) CD ll 17 then k- k+1) 1|. " I - eR1(F) + ... + e ‘Rk_1(i) + ekRk(F) + 0(e CDIO = w + eW1(F) + -.- + ek'1wk_1(F) + ekfik(F) + 0(ekl‘) and explicitly II Cl u(r,e,t) = I u (r,t)en£e, u Inlst n -n n v(r,e,t) = X vn(r,t)eMe, v lnlsMk l < -n n where . t . .un(r,t) = e'antfc - I e‘nwsan(r,s)ds], O < [nl s Nk fl 0 t D _ -4n t tn 5 vn(r,t) - e w [Dn - I e w bn(r,s)ds], O < lnl s Mk where c = meanIJteinwsa (r s)ds] O < lnl s N n 't 0 n ’ k t inws Dn = meanEI e bn(r,s)ds] O < In] 5 Nk t and t- u0(r,t) = [0Rk(r) - a0(r,s)ds t- v0(r,t) = [0Wk(r) - b0(r,s)ds. 18 1.3. Higher Dimensional Considerations. Consider the n-dimensional system in cylindrical coordinates (r.e.y) given by 9 = By + e”g(r.e.y.t.e)y + ekh(r.e.t.e) (3.1) i = eR(r,B,y,t,c) 6 = w + eW(r,O,y,t,e) where v and k are positive integers, B is an (n-Z) x (n-2) matrix with no pure imaginary eigenvalues and all functions are Zn-periodic in 9, almost periodic or P-periodic in t, and smooth enough for the following computations to be carried out. Let (re(t),e€(t),y€(t)) be a solution of (3.1) for which y€(t) and r€(t) are bounded, then |y6(t)| = 0(eL) for some L 2 O uniformly in t. Then decomposing y as y = (ys,yu) corresponding to the subspaces where B is stable or unstable. Then it is clear that t s ygm =- [ eB ‘t's’(evg(re.e€.y€.s.e)y€(s) + ekh(r€.e€.s.e))ds u.” u . y:(t) = f:e3 (t's)(evg(r€.e€.y€.s.e)y€(5) + ekh(r€.6€.s.e))ds where By = (Bsys,Buyu) where BS is a stable matrix and Bu is unstable. Then it is clear that lye(t)| = 0(eV*L) + 0(ek). 19 Thus L = k and lye] = O(ek). We have proved the following. Theorem 1.3.1. If (r(t),6(t),y(t)) is a solution of (3.1) with r(t) and y(t) bounded then [y(t)] = 0(ek). Thus if k in (3.1) is large enough one can essentially ignore the presence of y in the F and G equations. Since in this case we have eR(r,e,O,t,e) + 0(ek*‘) 10 II (D. II w + ew(r.e.o.t.e) + 0(ek+‘) k by proceeding and these equations can be averaged to the order of e as in section 1.2. The following theorem shows that a change of coordinates y +.y can be made so that in the new coordinates, k in (3.1) is as large as we wish. Theorem 1.3.2. Consider (3.1). There exists a function U = U(r,e,t,e) having the same periodicity properties as h(r,6,t,e) in e and t so that if y = y + ekU(r,9,t,e) then By + em§(r.e.§.t.e)§ + ek+‘fi(r.e.t.e) 9 R(P.6.§.t.8) ‘1. II (D + €N(P.9,S’,t.€) (D. II where g, h, R, W have the same periodicity properties in e and t as g, h, R, W respectively, and m = minIv,k+1}, further U is the 20 unique bounded solution of 3U 8U _ h + BU - w §§-- OED- 0 Proof. First decompose y = (ys,yu) corresponding to the subspaces where B is stable and unstable respectively. Then By = (BSyS,Buyu) where B5 and Bu are respectively stable and unstable matrices. (3.1) is then written as is = Bsys + evgllys + 8vglzyu + Ekhl 9a = Buyu + Ev9213’s + 8V922yu + 8khi r = eR(r,6,yS,yu,t,e) CD. II (D + 8W(Y‘,6,.Yssyu:t:€) where F 1 r- 1 911 912 Vs g(r.e.y.t.e)y = _ L921 9221 Lqu d an Eh“ h(r.6.t.8) = .hZJ with glj = 9ij(r’e’ys’yu’t’8) hi = hi(r,9,t,e). Here the dimensions are determined by those of y5 and yu. 21 Now let yS ' eku(r,9,t,e) ‘< m N yU ' ekv(r,6,t,e). ‘< n Then not writing dependence_on (r,e,t,e), we have _ - k - k - k 911(ys.yu)ys - 911(ys + e u. yu + e V)(yS + e U) k k 911(y5 + 6k“: in + EkV)yS + 5 [911(5kus E V) + G(U.V.§S.§u.6)§5 + H(U.V.§s.§u.2)§u]u def A - _ _ A _ _ _ k = 1911(ys’yu)ys + 2911(ys’yu)yu + e 611' Similarly 912(ys’yu)yu = 1612(95’yu)98 + 2612(ys’yu)yu + €kG12 k 921(ys.yu)ys = 1921(ys.yu)ys + 2921(ys.yu)yu + e 621 kc 922(ys’yu)yu = 1622(ys’yu)ys l 2622(ys’yu)yu l 8 22' Also R(ys.yu) = R(eu.eV) + R1(Is.§u)§s + R2(ys.yu)§u def x x x = R + R105 .ms + regs/5.9m W(ys.yu) = W + W1(ys.yu)yS + W2(ys.yu)yu. 22 Now we have 3 . ck au - Bu - Bu ys ' ys ' SE8r r +‘53'9 + BE' 3 s - k v A A - k B (ys + 5 U) T e E1911’s * 2911yu + 3 G11J v A — k k au au +€Efln% 2%flu+€hfl+€[h'wae'§fl k+1 Bu k+1 8U - e (R + R1yS + Rzyu) - e (W + les + Wzyu) k+1g k au BuJ _ 5- 111A - ___ _ ___ - B ys + e 91y + e 1 + e [11.l + B5 u - w 36 at where y = (ys.yu) min{v, k+1} a ll 9] = 9] (7399351598) 3") ll momma» Similarly, one obtains 3v ck u 3v €[h2+3v-w§§-fi]. yu = BU?u + emazy + €k+1hz+ To obtain the desired result the following equations must be SOIVEd. 5 Eu Bu _ h1+BU-wE-'a—'-0 3v av _ 23 The bounded solutions are easily seen to be t u(r,e,t,e) = I eBS(t-S)h1(r,w(s-t) + O,S,e)ds D“ v(r,e,t,e) = Ime3u(t's)h2(r,w(s-t) + e,s,e)ds. t Since h1 and h2 are 2n-periodic in a, so are u and v. We also have t+P s u(r,e,t+P,e) = I 1(r,m(s-t-P) + e,s,e)ds .00 t I eBs(t'o)h](r,w(o-t) + 6,o+P,e)do -oo u(r,6,t,e). The last equality holds since h], is P-periodic in t. If h.l is almost periodic in t, let {Tj} be a sequence so that h](r,e,t+rj,e) - h(r,9,t,e) + O as j +~w, uniformly in (r,6,t,e). Now one has t+Tj Bs(t+T-'S) u(r,6,t+rj,e) = I e J h1(r,w(s-t-rj) + 6,5,e)ds cum t S = I eB (t'°)h](r,w(O-t) + 9’ U + Tj,€)d0 .m t s eB (t-O)h1(r,w(o-t) + e,o,e)do .oo j-xao r1 = u(r,e,t,e). 24 Thus u is almost periodic in t with mEuJ c: mthll. A similar argument applied to v establishes that v has the same periodicity properties in e and t as hz. Then letting §(r.6.§.t.e)§ = (9.9.9231 g(r.6,t.e) = (111,112)‘ R(r,6,y,t,e) = R + filys + Rzyu W(r,e,§,t,e) = N + les + Uzyu U(r,e,t,e) = (u,v)' the theorem is established. 2. APPLICATION TO BIFURCATION PROBLEMS 2.1. Hopf Bifurcation for an O.D.E. in R2 Let us consider the nonautonomous O.D.E. in R2 2 (1.1) x f(x,t,a), x e R , t e R. a 6 (-a0, do) where f(O,t,a) s O and f is P-periodic or almost periodic on t. Suppose that the linear part of the equation linearized about x = O at a = O is independent of t and has the pure imaginary eigen- values 3 two, with “0 real and non zero. Then expanding f(x,t,a) in powers of x and a (1.1) can be written as (1.2) x = Ax + aB(t,a)x + G(x,t,a) where |G(x,t,a)| = O(|x|2) uniformly in t and a as |x| + O and B(t,a), G(x,t,a) are P-periodic or almost periodic in t. By the change of coordinates x + Px, where P is an appropriate 2 x 2 matrix we can assume that A is in Jordan form 0 -w A = 0 mo 0 Now write x G (x ,x ,t,a) x= 1 . G(X.t.a)= 1 1 2 x2 62(x1,x2,t,a) where for j = 1,2 we have 25 26 an (1.3) Gj(x1,x2,t,a) = kZZ Bj,k(x],x2,t,a) k n m B. (x ,x ,t,a) = Z bj’ (t,o)x x J,k l 2 n+m=k n,m l 2 n,m:O An infinite sum is indicated here only for convenience. Since only a finite number of terms will be considered (1.3) may be viewed as a finite Taylor development with remainder. Passing to polar coordinates in (1.2) by letting x = rNe, yields r = aNeB(t,a)Ner + cos 6 G1 + sin a G2 (1.4) e = mo + aTéB(t,a)Ne + %-(cos 9 62 - sin e 61) where for j = 1,2 G. = Z rk Z bJ’k(t,a)cos"e sinme J k=2 n+m=k n,m n,m:O dgf Z rk Z 8.] ,k(t ,a)ena(.e k=2 n=k(mod 2) " Inlsk with Bg’k(t,a) linear combinations of the bg’;(t,a) with complex coefficients and satisfying afak(t,a) = Bg’E(t,a) . Then 27 cos 9 G1 + sin e 62 = Z k 2 n=k(mod 2) Inlsk Bl’ -t’(e‘e - e'1'e)8:’k}eme “MB 1. 2 e£(n- l)B rk {ei(n+l)e 1 2 k 2 n=k(mod 2) Inlsk (s;’k - k) + (a; + tsfi’k)} "MB 2- rky,k+1(tae)nj'9 Z 2 n=k+l(mod 2) Y" Inlsk+l NI—l def ” k Similarly, we have cos a G2 - sin a G1 = 1 Q 2 =2 n=k(mod 2) n IN:k E rk 6k+1(tw)e k=2 n=k+l(mod 2) 6" lnl_<_k+l def ” k - kgz r Dk+1(6, t,a). Since both these expressions are real we must have YEn = Y:’ 55" = 65. k+ +1 and also note that LYk+1= L(Bk+l' £Bk+l)= dkil so that we have . k k . £¥k = 6k , -LYEk = 65k . 28 Further note that Ck and Dk are homogenous trigonometric poly- nomials of degree k with coefficients depending on a and t, P-periodic or almost periodic in t. Inserting these expressions into (1.4) we obtain (1.5) Then scaling by .where Ck and tions in powers (1.6) where h.o.t. = 2 3 orC2 + r C3 + r C4 +... 2 mo + aDZ + r03 + r D4 +... r + er, a + co in (1.5) yields a(arCZ + r2C3) + e2r3C4 + C(83) m0 + a(otD2 + r03) + ezrzD4 + 0(83) Dk are functions of (6,t,ea). Expanding all func- of so yields 2 3 _ 2 - a(arC2 + r C3) + e r C4 + h.o.t. = w + 8(aD + rD ) + eerD + h o t O 2 3 4 ‘ ° ° 0(23) + 0(52a), and Ck, Dk are evaluated at (e,t,0). We are now ready to average the e and e2 terms. Let 1| I 6 (DI ll - r + u](r,6,t,a) + c 2u2(r,e,t,a) + ev](r,6,t.o) + €2v2(r,e,t,a) with inverses satisfying r = r CD II CD. - eu](r,§,t,a) + 0(82) - ev1(r,§,t,a) + 0(82) 29 Then ; , au1 , au1 ; au1 r=r*€mrr+mr9+mr 4 SU . 3U . 3U *E‘m;r+fi§9+m% (1.7) g , av1 , av1 , av1 9=eiemrrifireimr 3v 3v . 3v 2 2 - 2 2 *8 a r+w Star In terms of the new coordinates F and 5, we have 2 3 F 10 ll - -2 storCZ + r C3] + 5 C4 2 3 8 - -2 - e (u -:-+ v -:)(arC + r C ) + h.o.t. 1 8r 1 36 2 3 2 2 F D CD. ll m0 + efotD2 + r03] + c 4 2 - - 6 (U1 §:-+ v1 %E)(a02 + r03) + h.o. . 3r ('1' where the right hand side is evaluated at F and 5. Also neglecting dependence on t and a, we have au‘( ) w‘ () -- r,6 = -:—'+ 0 c Br Br 3U au au 5‘61”.” = T]- ' €(U1‘a‘:+ V] 8—.) _..L+ C(82) 36 Sr 36 36 3U 3U 3U .__l = __l.- §_. §_. __l. 2 at (r,e) at 8(“1 a? + ”1 36) at + 0(5 ) where again the right hand side is evaluated at F and 6. Similar 3O expressions hold for auz/ar, auZ/ae, auZ/at, 3v1/ar, avI/ae, av1/at, avzlar, avz/ae and avz/at. Inserting these expressions into (1.7) and dropping the bars yields . 2 au.l Bu1 Bu.I Bu2 Buz Bu 2 1 c + r -——-o + w -——-+ -——- 2 3 + E [” c4 + ” 3r 3 as 3 o as at 3U U 2 2.. 2 __1 _1_ ' E (”1 8r + v1 ae)(arcz + ” c3 + ”0 as + at ) + h.o.t. (1'8) , 3vI av1 e=w0+EEGDZ+rD3+w03—O—+BTZ—J . av av 3v 2 2 2 __1_ :1 2 __2_ + e [r D4 + r 3r C3 D3 + “0— 39 + at 3- av av 2 .3. .L __1_ _L ' E ("1 3r + v1 ae)(°Dz + rD3 + “0 36 I at ) + h.o.t. First choose u1 and v1 so that the coefficient of e in (1.8) is independent of e and t. The following equations must be solved an au 2 1 l = 2 dgf arC2 + r C3 + mo 5§-+ 5E—' mgagtarcz + r C3] K1(r,a) (1.9) 3v 3v 1 1 - def 0102 + r03 + mo 5§—-+ 5t_" meanEaD2 + r03] - L](r,a) . 6,t Now C3 and D3 being homogenous trignometric in e of odd degree (i.e. 3) have mean value zero, thus 31 K](r,a) = or mean [C2] dgf arK1 6,1: L1(r,a) = a mean [Dz] dgf aL1 . 9.1; Then Theorem 1.2.2 implies that (1.9) can be solved so that u1 and v.‘ have the same periodicity properties as C2, C3, Dz, D3 in e and t if and only if the following integrals are bounded t tnw S I]: I e 0 (73(5), 63(5))ds for |n| = 1,3 0 n n I . It 21.1005 2. 0 e (y§(s). a§(s))ds t 13: (o (y§(s) - K], 53(5) - L1)ds where 6:(t)eme C = Z yk(t)en£6, D 3 Z k(mod 2) k n=k(mod 2) " k n Inlsk I Z and (u,(r,e,t.a).v,(r.e.t.a)) = (f2U1,3(9,t). rv,,3(e.t)) + 0(a) where (U1’3(e.t).v1,3(6.t)) = In)<3 (0?,3(t).v?’3(t))en£6 . $0 (1.8) can now be written as 32 Bu ° _ 2 3 3 3 1 3 r - eotrK.l + e [r C4 + 2r u1’3C3 + r 55-‘- DB au au 2 2 (1.10) O = w + e L + EZErzD + rzv C + r2 321-3-0 0 a 1 4 1,3 3 as 3 + w Ezg-+ 3X23 + h o t O 36 8t ° ° ° Now u2 and v2 must be chosen. The coefficient of 22 in (1.10) may be written as 3 2 3 3 (r h6(69t9a)9 r 96(estsa)) + “0 36'(u2:V2) + 5E'(U29V2) where 4 4 n26 (h ,9 ) = Z ( 96 )e 6 6 (")309234 n n 3 . 3 n 3 . 3 n £(n+k)6 + lnlzlkl=1 3((2Yk + 4n6k)u1,3a(Yk + LHOk)V]’3)e dgf I (hn’ gg)e“‘9 . |n|=O,2,4,6 The following equations must be solved Bu au 3 = 2 2 = 3 dgf 3 r h6 m0 5§—-+ 5t—' mgag [r h6] r K2 (1.11) av 3v 2 2 2 2 2 dgf 2 r 96 + “0 as + at ”gag [I 963 ’ L2 ' Again Theorem 1.2.2 implies that (1.11) can be solved so that u2 and v2 have the desired periodicity properties in e and t if and 33 only if the following integrals are bounded. t inwos n n 14: Io e (h6(s), 96(5))ds InI = 2,4 I ~ t (ho(s) - K 0(s) - L )d5 5' o 6 2’ 96 _ 2 t inw S , O n n _ 16. I0 e (h6(s), 95(5))d5 InI - 5 . Now for n = 6 in 16 we have 6 6 _ 3 . 3 3 3 . 3 3 (ha. 95) - ((213 + 3263)u],3. (Y3 + 3463)v1’3) -3£m t t 32m 5 (u?,3,v?’3) = e 0 [(c],c2) - I0 e 0 (y§(s),3g(s))ds1 where (c1,c2) are appropriately chosen constants. Then using the fact that Ly; = 63 this integral may be written as t Siwos 3 2 t Bimos 3 [I e 13(s1ds3 (1.22) - I e 13(s)ds (c1.2c2) O O which is bounded because I1 is assumed to be bounded. Since n = -6 in I6 is just the complex conjugate of this integral, I6 is always bounded. In the case where f(x,t,a) in (1.1) is P-periodic in t then the boundedness of I1 - I5 is equivalent to nwoP # 2nk for all integers k with InI = 3,4 (InI = 1,2 is redundant). If I1 - 15 are bounded and u],u2,v1,v2 are chosen accord- ing to Theorem 1.2.2, (1.6) becomes 34 1 o I _ 2 3 earK1 + e r K2 + h.o.t. (1.12) mo + eoLL.l + ezrzL2 + h.o.t. a. II The following theorem summarizes the above results. Theorem 2.1.1. Consider the differential equation cEarC2 + r2C3] + ezr3C4 + h.o.t. - _ 2 2 O - mo + etaD2 + r03] + e r D4 + h.o.t. ‘10 II where h.o.t. 0(53) + 0(cza) uniformly in e and t as IeI + IaI + 0. m0 f O is real. Assume that Ck and Dk are - . - . . g 2 related by Ck - NeGk’ Dk - TeGk with Gk Gk(e,t) E R a homogenous trignometric polynomial of degree k-l with coefficients_ almost periodic (P-periodic) in t. Then there exist functions u1,v],u2,v2 which are Zn-periodic in 9, almost periodic in t so that if r = F + cu + ezu ' 6 = O + ev + c v 1 2’ 1 2 and the integrals I1 - 15 are bounded (nwoP # 2nk for n = 3,4 and all integers k, if the functions are P-periodic in t) then s _ - 2-3 3 _ 2-2 where 35 K = mean [C J , L = mean [D J l e,t 2 1 e,t 2 K2 = mean [C4 + auC3 +— 36 D3] = Lv L2 mean [D4 + vC3 + 36 D3] with u and v defined by Lu+ Bu _ C3 + “0— 39 51" 0 Lv 3v = D3 + “0 36 '5? 0 ' If K1 - K2 < O the choice of a = 5 brings (1.12) to 10 I ' eZErK1 + r§K21 + 0(53) 2 2 3 - m0 + 8 [L1 + r L2] + 0(e ). (Do I So the presence of a "cylinder like" invariant manifold with radius approximately: is suggested. The existence, uniqueness and periodicity properties of this structure are proved in 2.3, in a more general setting and are not treated here. If r = r(e,t,a) defines an invariant manifold of (1.1), 2n-periodic in 9, almost periodic (P-periodic) in t, which bifurcates from r = O at a = 0, we need to show that if (r*(t),e*(t)) is any solution lying on this manifold that this 36 solution satisfies (l.l2) (i.e. no manifolds of the desired type are lost in scaling). Since we scaled by r + er, a + so and then took a = e, we must show that 1. l 1m E-r(e,t,a) 6+0 exists and is finite. Because of the periodicity properties of r(e,t,a) in 6 and t its gradient must vanish some place (see Appendix), say Vr(60,t0,a) = (0,0) . Suppose that * 'k - (r (t0)99 (t0)) ' (r(eostosa)seo) then by scaling by e = r*(to)/r0 we can assume that r*(t0) = r0 and since 0* 0* r (to) = vr(609t0:a)(e (to): 1) = 0 we have 0 = eroK](a - e + 0(ea) + C(82)) . Where upon scaling again by a + ea gives a = l + 0(a). This scaling brings (l.l2) to 2 - e rK](l - r0 '3. II 2r2) + C(63) + C(52) . CDO ll mo Now consider the thin cylindrical shell given by 37 dgf def C: r1 (1 - v)r0 §_r 5_(l + y)r0 = r2 where for appropriate 7 + 0 as e + 0 we have at r = rj, j = l,2, if e is small enough +192 = e4r1r2K§(i - (l - y)2)(l - (l + Y)2) + 0(85) < 0 and sgn(r]) = sgn(K1) . Thus the cylindrical shell, C, is positively invariant if K1 > 0 and negatively invariant if K1 < 0. In unscaled coordinates we have er1 5_r(e,t,a) 5_er2 so that lim g-r(6,t,a) = r0 , a = 82 + C(83) . 5+0 Hence no manifold of the desired type is lost in scaling. Theorem 2.l.2. Suppose that the integrals in I1 - 15 are bounded, (nmoP is not an integer multiple of Zn for n = 3,4 and all integers k, if f in (l.l) is P—periodic in t). Let K1, K2 be defined in (1.9) and (l.ll). Suppose K1 - K2 < 0. Then if r = r(e,t,a) defines an invariant manifold for (l.l) bifurcating from r = 0 at a = 0, which is 2n-periodic in 6 and almost periodic (P-periodic) in t, then any solution lying on this mani- fold may be obtained by the scaling r + er, a'+ so and averaging so that in the scaled and averaged coordinates (l.l) becomes 38 '1 o I ' earK] + €2r3K2 + C(83) mo + eaL1 + Ezrsz + C(83) . 6 Then letting a = 5, r0 = (-K1 - K51)15 so that a 52 (unscaled) and lim JE- r(6,t,a) = r0 (unscaled). 8+0 If K1 - K2 > 0 the same result holds except we now let a = -e, r0 = (K1 - K51)15 so that a = _€2 (unscaled). 2.2. Higher Dimensional Hopf Bifurcation Consider the O.D.E. (2.l) 2 = f(z,t,a), z E R", n 3_3 where f(0,t,a) 5 0, f is almost periodic (P-periodic) in t uni- formly for z and a in compact sets. Further suppose that gg-(O,t,0) dgf A = independent of t and that A has the pair of pure imaginary eigenvalues :_iwo. with wo real and non zero and all other eigenvalues of A have non zero real parts. Then (2.l) can be written as (2.2) i = A2 + aB(t,a)z + F(z,t,a) where |F(z,t,a)| = 0(Izl2) uniformly in t and a as |z| + O, and B(t,a), F(z,t,a) are almost periodic (P-periodic) in t. By the change of variable 2 + Pz where P is an n x n matrix we can assume that A is in the form 39 and AQ has no eigenvalues with zero real part. 2 n-2 Now let 2 = (x,y) e R x R and 2 n-2 Hacm=<fiu¢axrgunmieaxa F311(t.a) 312(ts0f B(t,a) = 821(t,a) 822(t,ai‘ L where B]](t,a) is a 2 x 2 matrix. Then (l.2) becomes x = Apx + aB]1(t,a)x + a812(t,a)y + F1(x,y,t,a) (2.3) 51 = AQy + aBz](t,a)x + a322(t,a)y + F2(x,y,t,a) where lFi(x,y,t,a)| = 0((le + |Y|)2)’ i = 1,2. Then expanding F1 and F2 in powers of x and y, we have F1(x.y.t.a) = F§’°(t.a)x2 + F}"(t,a)xy + F$’2(t,a)y2 +... F2(x.y.t,a) = F§’°(t,a)x2 + Fg"(t,a)xy + Fg’2(t,a)y2 +... where the notation indicates that F?’2(t,a)y2 is a bilinear map n-2 n-2 xR mm W w oph)+£¢uauhmg.mdm 2 from R have let F?’2(t,a)(y,y) = F?’2(t,a)y with similar interpretations for Fg’m(t,a)x"ym. Then passing to polar coordinates in x by setting x = rNe, (2.3) becomes 40 ‘30 I ‘ aC2(6,t,a)r + C3(B,t,a)r2 + C4(B,t,a)r3 + (ac1(e.t.a) + 62(e.t.a)r + 63(e.t.a)r2)y +m#)+mwfi) (2.4) 6 mo + aDZ(B,t,a) + D3(B,t,a)r + D4(B,t,a)r2 + r-](aD](B,t,a) + 52(6,t,a)r + 53(6.t.a)r2)y + 0(r3) + r'10(IYi2) AQy + a322(t,a)y + G(r,6,y,tsa)yz ‘(o II + GE](estsa)r + E2(estaa)r2 + 0(r3) where Ck, Dk’ Ek, Ck, 6k are homogenous trignometric polynomials of degree k with coefficients depending on t and a. Then scale by r + er, a + ea, y + ey in (l.4) to obtain r = cEarCz + r2C3J + crazy + 82r3c4 + ((lel + Ial + lyl)3) (2.5) B = mo + EEaDZ + r03] + efizy + €2r204 + 0((Iel + Ial + IYI)2) ' _ 2 2 y - AQy + eHy + etarE1 + r E2] + 0(e ) A where Hy = aBzzy + Gyz, and Ck’Dk’Ek’ek’Dk are evaluated at (B,t,ea). So that by expanding these functions in powers of so does not change the form of (2.5), we can assume that they are evaluated at (e,t,0). We are now in position to apply Theorem l.3.2 to decouple the r and 6 equations from the y equation up to the cubic order. Let y = y + eU where U = U(r,e,t,a) is the unique bounded 41 solution of 2 Bu Bu _ arE1+rE2+AQU-mO-a-é--§t--O. Then Theorem l.3.2 implies that (2.5) becomes 9 = etarCZ + r2C3] + EZErEZU + r3C4] + 0((lel + hxl+ I§I)3) (2 6) é = w + EEaD + r0 1 + 52:6 u + r20 J + 0((lel + hx|+ l’l)3) ° 0 2 3 2 4 y : _ - -—- 2 y - AQy + eHy + 0(6 ) where H'= Hir,e,y,t,a,e). And Theorem l.3.l gives lyl = 0(62), so 0((lel + lal + [y|)3) = 0((|€l + Ia|)3). Further note that U = arV + rzw so that (1.6) can be written as r = eEarCz + r2C3] + e2r3cfizw 4 c4] + 0((lel + lal)3) (2 7) B = w + EEaD + r0 3 + eerEB W + D J + 0((lel + lal)3) ‘ ° 0 2 3 2 4 y = A0; + C(62) as long as r and 9 remain in a bounded region as e + 0 and r is considered to be away from 0. Theorem 2.2.l. Consider the differential equation in (r,B,y) 6 R x R x R"-2 9 = etarC2 + r2C3] + crazy + €2r3C4 + h.o.t. é = mo + EEaDZ + r03] + efizy + €2r204 + h.o.t. y = AQy + cHy + etarE] + r2E2] + C(82) where h.o.t. = 0((lel + lal + |y|)3), 0(52) are uniform in t and 42 e as Isl + Ial + lyl + 0 and all functions are 2n-periodic in 6, almost periodic (P-periodic) in t, AQ has no pure imaginary eigenvalues, ”0 f 0 is real, subscripted functions depend on (e,t), H = H(r,e,y,t,a,c). Then there exists a unique function U = U(r,9,t,a), 2n-periodic in 6 almost periodic (P-periodic) in t so that if y = y + eu then [9| = 0(e2) and 2 2 3 - eEarCz + r c3] + e r [62w + c4] + 0((|e| + |a|)3) '3 o I 6 = wo + eEaDz + r03] + ezrztfizw + 04] + 0((lel + la|)3) ‘ 0 then a = -e2 and a = 52pm1 + r3K2) + 0(a3) B = mo + 82(-L1 + rZLZ) + C(83) § = A0; + 0(22) . 2.3. Existence of the Manifold In the previous section it was shown that the system (2.1) after scaling and averaging can be written as r = €2(:_VK] T T3K2) * 0(83) (3.1) 6 = mo + 52(L1 + rsz) + 0(e3) y = AQy + 0(ez) where :_= -sgn(K1 - K2), K1 - K2 # O and A0 has no pure imaginary eigenvalues. We now prove that if e is small enough there is a unique two dimensional manifold parametrized by e and t, 2n-periodic in 6, almost periodic (P-periodic) in t so that solutions of (3.l) which begin on M will remain on M for all time. More generally the following theorem holds. 46 Theorem 2.3.l. Consider the differential equation in the n-2 coordinates (r,6.y) ER x R x R given by r = er + eaR(r,B,y,t,e) (3.2) B = w(e) + ebW(r,B,y,t,e) v = Ay + ecY(r.6.y.t,e) where R, N, Y are 2n-periodic in 6, almost periodic or P-periodic in t. A has no pure imaginary eigenvalues. All functions are continuously differentiable, a > l, b > l, c > 0. Then there exists an so so that if 0 < e 5-50 then there are unique func- tions r*(e,t,e), y*(e,t,e), which are 2n-periodic in 6, almost periodic (P-periodic) in t. So that for a fixed 5 in (0,803 the two dimensional manifold M defined by 'k 'k M: (ray) = (Y‘ (9,t,8), .Y (69t98)) is invariant under the flow induced by solutions of (3.2). 310233 DECONPOSE y '-' (yl 9y2) 6 Rk x Rz according to the subspaces of R"-2 where AQ is respectively stable and unstable. Let Asy1 denote Ay restricted to the stable subspace and Auy2 for Ay on the unstable subspace. Then (3.2) has the form (3.33) ;'= 8" + 8aR(r,e,y] ,y2,t,'€) (3.3o) é = w(€) + ebW(T,9sy1,y2,t,e) (3.3c) y1 = Asy1 + eCY](r,e,y1,y2,t,e) (3.3d) 9, - A”y2 + e°Y2(r,e.y1,y2.t.e) . 47 Define X to be the space of triples of functions k (f(e,t), g](B,t), 92(6,t)) taking values in R x R x R2, where k (yl, yz) E R x R9“, which are 21r-periodic in 9, almost periodic (P-periodic) in t. "f“: 5UP ”(9913”: 09'191“ i D: “9"” 5.09 e t ‘ lf(9,t) ' f(6;t)l f_6|9 ' 6]: l9](e.t) - 91(53t)l 5_ole - 6], Igz(eat) ‘ 92(§;t)l f,5|9 - 5] . X is clearly complete in the norm “-H. We will define a mapping F: X + X so that the unique fixed point of F in X will define an invariant manifold of (3.2) with the desired periodicity pro- perties. Let (f,g].92).€ X, and (r,y],y2) = (f,g],gz) in (3.3b). We have 6 = m(€) + Ebw(f(est)sesg](est)sgz(est)st9€) which has a unique solution passing through (Est) denoted by ‘k * B = 6 (t; Tog; f]:g]992) - * Substituting 6 for B, (f1,g],gz) for (r,y],y2) into the re- maining equations in (3.3) yields '10 ll er + eaR(f(t.e*).e*. g,(t.e*). 92(t.e*).t.e) * * * * Asy1 + ecv1(f(t.e ).e . 91(t.e ), 92(t,e ),t,e) c<. -—-l u 48 o _ u c * * * * yz - A yz + s Y2(f(t.6 ),e , 91(t,6 ), 92(t.6 ).t.e). Then the variation of constants formula gives the unique bounded solutions as Ea {t ea(s-t) J-oo * r(t; E; fag]:gz) R(f,e ,g],gz,s)ds t S 5c I 9A (t-S)Y](fse*:91a9295)d5 Q y](t§ 53 f:91s92) . . c A“(t-s) * y2(t’ g: fog]:92) '8 f: e Y2(f,9 ,gI,92,S)dS define F(f,g],gz) = (r(t,B), y](t,e), y2(t,e)). We will show that F: X + X is a contraction in ”'N- To accomplish this, first define Luz) = maxfllRll, 135:. ug—Si. 1:3—51-11, "37’2“, unfit} with similar definitions for L(N), L(Y]), L(Y2). Then t lr(t,6)| §_eaL(R) I e€(s't)dt = ea"L(R) t |y1(t,B)| 5_ecL(Y1) J leAs(t'S)|ds Aso -Ys° but there are constants Ys’ K so that le I 5.Kse , thus 5 |y1(t,6)| 5_eCKSy;1L(Y1), and similarly |y2(t,e)| 5.3cKuy;]L(Y2). Thus if e is small enough we have “r" §_D, fly1fl 5,0, HYZH 5.0- Now let (F.§I,§2) = (r(t,5). y1(t,§), y2(t,5)) then '(r9Y]9YZ) ‘ (F9y1’y2)l §.Ir ' izl + IY] ' 91' + lyz ' yzl 49 but a t a(s t) Ir - rl < e I e IR Rlds t y(S-t) C S ly1 - yll < 2 KS J_me |Y1 - Vilds _ Yu(t'5) lyz - yzl < e Ku t e |v2 - Yzlds where _. * _.*- .- IR ‘ RI = IR(f,9 $91992) ‘ R(?9e :91992)l :.(36L(R) + i>|e* - 6*! [v] - l3! §_(36L(Y1) + i)|o* - o’l _ 'k _* IY2 - Yzl £.(36L(Y2) + 1)I6 - 6 l and 'k * bt * B=6(LM=B+eJNUfi,%Qfl$+wkHt-fl J 'k _ T e = e (t,B) . Let M = max{L(R), L(N), L(Y]), L(Y2)} and B = 36M + 1. Then * _.* _. b t 'k _.* le - o I 5_|e - 9| + e B I la - e lldsl '1' so the Gronwall inequality yields * _ m -?|5m-omma%n-tn. Thus __ t Ir - Fl‘g eaBIB - elJ exp((e - ebB)(s - t))ds 50 since b > l choose 5 so small that e - ebB > D, then also - c —- t b Iy1- y1l se KSBIG - 6|] expihS - e Bus - mas - c - b lyz - yzl 5.€ KUBIe - e| texp((vu - e B)(t - 5))ds so that a must be small enough to have 75 - ebB > 0, Yu - ebB > 0 in which case _ eCKSB _ eCKuB |y1 - ynl §.--:-Bgu lyz -.y2|:s'-jj-1fi; 75 6 Yu 8 and Hray] 0’2) " (Fsy'l 9.92)] 5. Q(€)I9 ' 6" where c c a e K B e K B _ e B s u 0(8) ‘ o T b T “““13' s - e B Y5 - e B Yu - e B thus if e is small enough 0(a) < 6. Next (r,y1,y2) must be shown to have the same periodicity properties as the functions in X. If (f,g1,gz) are 2n-periodic in 6, then by uniqueness of solution we have 9*(t; T, g + 2n) = 9*(t;t,g) + 2n and since R, Y1, Y2 are Zn-periodic in e we must have (r,y1ay2) 2n-periodic in 6. 5] If (f,gI,gz) are P-periodic in t then again by unique- ness of solution we have * ' * 9 (t + P; T + Ps5) = 6 (t; T.€) and then t+P r(t + P.e) = 5a ( e€(s-t-P)R(f’e*’gl’92)ds .-oo * where f, g], 92 are evaluated at (B (s; t + T,e), s) so that after the change of variable 0 = s - P t e5(°'t)R(f.e*.g].92)ds r(t + P,B) = ea I * where now f, g], 92 are evaluated at (6 (o; t,B), o) and we have r(t + P,e) = r(t,B). Similarly y1, y2 are P-periodic in t. If the functions in X are almost periodic in t, let {Tj} be a sequence so that (f991sgz)(t + Tj) " (fag'l ng)(t) + 0 (R,H,Y1,Y2)(t + Tj) ' (R,N,Y1,Y2)(t) + 0 unifbrmly in t and the remaining variables as j +,w. Then since * *' B (t + Tj; T + rj,e) = e (t; 1,6) we have t+t. €(S-t-T.) J f * d r(t + Tjse) = 8a 8 R( ,6 991992) S .00 * where f, g], 92 are evaluated at (6 (s; t + Tj,9), s), after the change of variable 0 = s - Tj 52 t r(t + Tj,6) = ea I 66(O-t)R(f,B*,g1,gz)do coo where f, g], 92 are evalbated at (6*(0; 1.6), o + rj), passing to the limit we have established that r(t,e) is almost periodic in t. Similar arguments establish that y1(t,B) and y2(t,e) are also almost periodic. Thus (r,y1,y2) 6 X whenever (f,g],92) 6 X. so if e is small enough F: X + X . Finally, it is shown that F is a uniform contraction in the supremum norm on X. Let F = r(t,e; f,§],§2), similarly for y], 92. We have l(r.y].y2) - (r.§1.§2)l 5,Ir - Fl + ly1 - 9]! * lyz - 92l. Now t |r - Fl geaf e€(5"°)|R - R|ds ‘ cK t Y5(S t) Y T’ d lyl ‘ yll < E s e l l ' 1| 5 ' c Y“(t-S) Y Y d (Y2 ' y2| < 5 Ks te l 2 ' 2l 5 where _ * __*_ _ IR - RI = |R(f.e .91.92) - R(f.e .91,92)| .. - .. * _.* s.MCUf - f” + H9] - 91” + "92 - 92“] + MIG - e l 'k dgf MA + Mle - 6*l. 53 With the identical inequality holding for IW — W], lY1 - Vila |Y2 - Y2|. Then t * * §_chA|t-T| + ebM ( le - o ||ds| T and by the generalized Gronwall inequality A _* ' t b _ m*-e|_Jmn-t|+9fi[o€WS”p-tnml Jr b = [e8 "It‘Tl - lJA so that - ebMIt-II |R - R| §_MAe with identical inequalities holding for |Y1 - T}|, |Y2 -‘Té| which implies that t b a Ir _ Fl §_eaMAJ e(e-e M)(s-t)ds = a MA -°° e - ebM provided e - ebM > 0. Similarly if ys - ebM > o, yu - ebM > 0 we have _ ecMKSA _ ecMKuA Ys - e M y - e M U thus ((rsy'l 93,2) ’ (F3571 ’92)) i T(€)A where 54 c c - 8aM + e MKS + e MKu We) " "o ""' 'b b e - e M 75 - e M Yu - e M so that if e is small enough T(e) < l and F is a uniform con- traction on X. This completes the proof of the theorem. To apply Theorem (2.3.l) to (3.l) first suppose that K1 - K2 < D and let r = r0 + 5%? where r3 = -K1 - K5] so that h = -2€2K1? + 0(85/2) ' g 2 5/2 B mo + 8 (L1 + rOLZ) + 0(e ) y = AQy + 0(82) 2 and then replacing c with e, t with -2K1t, and dropping the hats yields r ' r + 0(65/4) o = w(€) + 0(a5/4) Y = AQy + 0(8) with w(€) = -2K](w + a(L1 + r0L2)) and Theorem 2.3.1 applies. 0 Similarly if K1 - K2 > 0. Also the cylindrical shell given by * (3-4) C : (l - y)r0 5_r §_(l + y)r0; lyl = 0(8) with y + 0 as e + D with be invariant only if AQ is a stable matrix and K1 - K2 < D with K1 > 0 (K1 - K2 > D with Kl < 0). In which case the manifold is stable. If AQ has at least one * eigenvalue with a positive real part then solutions may enter C 55 through r = (lg:y)r0 but will leave along the eigendirection of this eigenvalue. Theorem 2.3.2. Let K1, K2 be defined as in Theorem 2.2.2. If K1 - K2 f 0 then the system (l.l) of section 2.2 has a unique invariant manifold M defined by M: (r,y) = (r(9.t.e), Y(6.t,e)) where r and y are 2n-periodic in 6, almost periodic (P-periodic) in t, (r,y) +-(D,D) as e +-0 with 82 = esgn(K.l - K2)a. So that in the original coordinates (x,y) with x - (r cos B, r sin e) 2 -l » r(B,t,e) = ero + on: ), r0 =|K1- K v Y(9,t.e) = 0(6) . M is stable if and only if all the eigenvalues of AQ have nega- tive real parts and K2 < D. 2.4. An Example Consider the forced Van Der Pol equation .. 2-_ (4.l) x + x - a(l - x )x - f(t) where x e R, f(t) is almost periodic or P-periodic and t I N f(s)ds, N = (cos 9, sin 6)‘ 0 S B is bounded. Let u(t) be the unique almost periodic (P-periodic) solution of 56 (4.2) U + u = f(t) then ' t u(t) = Néc + J sin(t - s)f(s)ds . 0 And let y = x - u so that after using (4.2) we have (4 3) Y + y - 89(y.9.t) = 0 where g(y,y,t) = (l - u2 - 2uy - y2)(y + u). Writing (4.3) as a system in R2 yields (4.4) 2 = A2 + eF(z,t) where Y 0 l 0 Z = 9 s A = _] 0 9 F(Z,t) = g(Z,t) Passing to polar coordinates by setting 2 =‘rNe, gives 2 3 2 + r C3 + r C4] -l 2 -l - ctr D1 + 02 + rD3 + r D 1 o I ‘ EEC1 + rC (4.5) m. II 4] where C = (l - u2)u sin 6 l _ 2 . 2 - . C2 - (l - u )Sln B - 2uu Sln 6 cos a C _ ° 2 . . 2 3 - -u cos 9 Sln e - 2u Sln 9 cos a C4 = -sin26 c0528 (l - u2)u cos 9 O ...a II 57 D2 = (l - u2)sin 6 cos 6 - 2uu cosze 03 = -u c0536 - 2u c0526 sin 9 D4 = -cos3e sin e . So that (4.5) may be averaged provided the following integrals are bounded. t Lns A]: I e u(s)ds for Inl = 1,3 0 t tns 2 ' A2: I e u (s)ds for lnl = 2 . o t .n 3 A3: I e‘ Su (s)ds for |n| = l 0 t A4: I u2(s) - mean [uzlds . 0 t In which case the averaged form of (4.5) is - l 3 r = e[%-Kr - §-r J + 0(82) (4.6) B = -l + 0(82) where K = mean [1 - u2]. t Thus an invariant manifold near r0 = 2Kla is expected. In fact if r + r0 + egK'lr and t + -t then r = r + 0(83/2) 6 =1 +o(e2) and we may apply Theorem 2.3.l to assure the existence of a manifold 58 of the form r = r(6,t,e) = r0 + 0(a) which is invariant under the flow induced by solutions of (4.4) and r is 2n-periodic in 9, almost periodic (P-periodic) in t. If f(t) can be written in a finite Fourier series - v - f(t) - X ave , X0 - 0 u then if leI f l for all v A t u(t) = E uve V , "v = av(l - XS) 1 so that A1 - A4 are bounded if n + Xi f 0 fOr |n| = l,3 (4.7) n + xi + Xk f D for [n] = 2 n + Xj + Xk + X2 f D fbr [n] = l 3. FUNCTIONAL DIFFERENTIAL EQUATIONS 3.1. The Abstract Equation. Consider the retarded functional differential equation (RFDE) (1.1) 2(t) = f(zt,t,a) where def 2t 6 c ... C([-r,0], R") zt(e) = z(t + e) for B E [-r,0] and f: C x R x R + Rn is almost periodic (P-periodic) in t and smooth enough for the following calculations to be carried out. Assume further that f(0,t,a) = 0, 3f 0 t’0 = L th where L: C + Rn is bounded, linear and independent of t. Thus L has the Stieltjes integral representation 0 w=( mono) '7‘ where n(e) is an n x n matrix function of bounded variation. Then (l.l) becomes 2(t) = th + H(zt,t,a) (l.2) HWJa)=WHmM+FMJ&M2 59 60 where as before F(¢.t,a): C x C +-Rn is a symmetric bilinear map. Upon scaling by z + £2, a + ea (1.2) becomes (with a different H) (l.3) 2(t) = th + eH(zt,t,a,e). Now v(t,-) 6 C is a solution of (l.3) if and only if v(t,6) = zt(6) where z(t) satisfies (l.3). This fact gives us the clue as to how (1.3) can be rewritten as an ODE in an appropriate Banach space. 1 Lemma 4.l.l. If v(t,e) is C in t E R and e 6 [-r,0] then a necessary and sufficient condition that v(t,e) = u(t + e) l . av _.§1 15 that 5-6-- at . for some u in C 3399:, Necessity is obvious. 0n the other hand along 6 + t = a we have é%-v(t,e) = D so define u(a) = v(0,a). Thus v(t,e) is a c1 solution of (1.3) if and only if av(t,B) = 3v(t,e) 3t 36 (l.4) é%-v(t,0) = Lv(t,-) + eH(V(t,-).t,a,8)~ I Now define E = C e Rn and A: C .+ E by v +r(v,Lv - 9(0)). Then (l.4) may be written as (1.5) git- (v(t.e),o) = Av(t.-) + (o.eH). Now suppose that A has a pair of pure imaginary simple eigenvalues tiwo, wo # D and all other eigenvalues of A have nonzero real parts. It is well known that all eigenvalues of A 61 are isolated and of finite multiplicity and are determined by solving 0 X6 det(XI - I dn(e)e ) = o -r Let C* = C([D,r]. R") (row vectors), E* = C* o R". Define a bilinear form on E* x E by ((wsa)9(¢9b)> = a°b + a°¢(0) + W(O)°b + [W9¢J 6 time] = v(0)-¢(0) - (O I 4(5 - 6)dn(e)¢(5)ds -r 0 so that A*, the adjoint of A, is given by A* = i-o.L*o - l> ,, o Lw=( Mammo. -T‘ Let e = (¢],¢2) be a basis for P = N(A:£w01) and W = (41,42) a basis for N(A*tim0I) chosen so that [W,¢l = I. Then any (v,b) 6 E can be written as (v,b) = (VP.0) + (vQ.b) vP ¢EW,VJ Q _ P V -V-V. Note that [W,vQ] = 0 and E = P 9 Q. Also there is a unique 2 x 2 matrix Ap so that Ao = oAp and Ap has eigenvalues :Lwo. 62 Now for any (v(t,e),b) e E we have wuehm=lfuekm+(D)Ng 3 62¢ = anNé?(0)[¢(-l)¢(0) + a(0)o(-l)mE E = ane(-l)N€e(0)Ng(-¢v(0),l) = (a + BfItNTéW(0)¢(-l)Ng o 3 = anTéW(D)¢(-1)Ng¢(0)Ng and note that only C2 and D2 depend on t, so that the averaging can be carried out if 12 and I3 (section 2.1) are bounded (kan is not an integer multiple of Zn for |k| = 2). The averaged form of (2.5) is i = -eK](a,6)r - eszr3 + h.o.t (2.6) é = -on - EL](a,5) - €2L2r2 + h.o.t d ‘ - A + 0 + + 0( 2) d—t-yt _ Qyt (€(a 5)) 8 where K](a,6) = meanEC2] €.t 65 _ -I A (2.7) K2 - meanEbn C303 + C4] 5 where C4 = CZW and 'W = (W,D) is the unique Zn-periodic in a solution of 3N ._ (2.8) E + A0“ + bn 5:... 0 with similar expressions holding for L](a,6) and L2 and Iyt ' ytl = 0(8)- Now straightforward computation yields (2.9) K1(a,6) = "2 (a + <5mean[f]) l+a t n (2.lD) magnEC3D3] = 0. To compute K2, write E E1 + Ez where E1 = -%bnsin2§-(O,l) I (o,hk)ek‘5 I=2 |k :2 = -eonsin2t<¢(-l))e‘kg |k|=0,2 with Y4k = yk, 8-k = Bk, so that K2 = mignfczW] = 2Re(y_2w2(0) + B_2w2(-l)) y = bn(bn-i) '2 2(l+ag) an(l+ibn) -———-—7f—— . 2(l+an) Now wk(e) = uk(e) + v(e) corresponding to W = W1 + W2 so that (2.ll) for j = 1 becomes (0.0) = Z (fik(e) + kibnuk(6). hk - anuk(-l) - ak(0))ek‘5. |k| 2 Hence . 2th 6 _ - 2+4 -l n "2(6) _ J—fi—i). e and then (l ) A a " a g 20(l+an) For j = 2 in (2.ll) we have (0.0) = Ik%=2(ik(e) + kzbnvk(0) + 9k(6). -anvk(-l> - ikiollek‘E. which yields -2£bne v2(e) = e [v2(0) - P(e)] 67 where e 241bns P(9) = I e g2(s)ds 0 02(0) = (an + zzbn)(anP(-l) - 02(0)). Explicitly it is found that P(-l) = -——l—§-[2(an - l) - z(on + (-l)")l 6(l+an) ( Ib" 9 0) = 2 204$ so that -1 , v (0) - (2 + 4b ) 2 6 «l+a2) n n l . n v (-l) = (2a - L(-I) ) 2 on”? " and then meanECszl = D. 6 Thus we have a (l-3a ) RAE K2=JL—%L D, (2.1) will have an invariant mani- fold with amplitude approximately r0 where t (2.14) r0 = (aM1 + 6M2) . Since an is given by (2.2), if n is even an 2 n/2 so that M1 < 0 and MI > 0 if n is odd (see Figure 1). Now by Theorem A5 in Hale [8] all roots of the characteristic equation (2(t) = -az(t - 1)) (2.15) . AeA + a = 0 have negative real parts if and only if D < a < n/2. Since for a = a0 = n/Z, X = tin/2 are the only pure imaginary roots of (2.15) and these are not eigenvalues of A0, we must have all eigenvalues of AQ with negative real parts. For a = an, n f 0, AQ must have eigenvalues with nonnegative real parts. Thus, only the manifold bifurcating at (a,6) = (n/2,0) is stable. 69 )2 '7 “"‘ * 5t;<\\\\\\ \\ a = -6mean[f] t 1 "1 > 0, (n even). a = -6mean[fl t m MN I M < O, (n odd). Figure l. Bifurcation diagrams for equation (2.1). APPENDIX 70 APPENDIX In this appendix the basic theory of almost periodic functions is outlined. No proofs are given for the classical results as they are readily available in standard texts on the subject, for example in Bohr [2], Favard [7], Hale [9]. Definition 1. A continuous f: Ri+ R (or C) is said to be almost periodic if given e > 0 there exist 2 = 2(3) > 0 so that for all a 6 R there is a T E Ea, a +.£] with |f(t + T) - f(tM < e for all t e R. r is called an almost period of f in Ea, a + £1 relative to- e. ' A1. If f(t) and g(t) are almost periodic then so are f(t + a), f(at) for a real, zf(t) for 2 complex, f"(t) for n = o,l,2,3,---, |f(t)|, f(t), f(—t)-+ g(t), f(t)-g(t). In fact, if F: R2 +YR (or C) is uniformly continuous then F(f(t),g(t)) is almost periodic. _ A2. If f(t) is almost periodic then so is f'(t) provided it is uniformly continuous. A3. If f(t) is almost periodic then (tf(s)ds is almost periodic if and only if it is bounded. A4. If f(t) is an almost periodic function then the fol- lowing limit exists, is finite and independent of a e R. T lim %-[ f(t)dt de meanEf]. T+w a t 71 A5. If f(t) is almost periodic then there are at most a countable number of X so that a(X) dgf meanEEAf] # 0 where t Ex(t) = exp(LXt). Any X for which a(X) # D we call a frequency of f with Fourier coefficient a(X). If {An} is a sequence of real numbers and Zran = D, where rn is an integer implies that rn = 0 for all n, then "‘ {Xn} is said to be rationally independent. The span of {An}(sp(Xn)) is the set of all linear combinations of the X" with integer coefficients. If {An} is rationally independent and {an} c:sp(Xn) then {An} is called a basis for {an} c:R. Definition 2. If f(t) is almost periodic with frequencies {kn} then the module of f, mEf] dgf sp(Xn). If {An} has a finite base then f(t) is called quasi-periodic. The following result is very useful in showing that a function is almost periodic. A6. If f(t) is almost periodic and g: Ri+ R (or C) and for any sequence {Tj} c:R with f(t + rj) - f(t) +sD uniformly in t as j +~m we have g(t + Tj) - g(t) + 0 uniformly in t as j +~m then g(t) is almost periodic and mtg] c:m[f]. Definition 3. A continuous function f(x,t) 6 RH (or C") is said to be almost periodic in t uniformly with respect to x in compact sets if given a compact set K c:Rn and e > 0 there exists 2 = £(e,K) so that for any a 6 R there is a T 6 Ca, a + £1 with If(x,t + T) - f(x,t)l< e for all t e R and x e K. In what follows and in the text we will refer to such functions simply as almost periodic in t. 72 Now if f(e,t) is 21-periodic in e and almost periodic in t then we define 1 2n meanEf] = 55-] meanEf](B)d9. B,t 0 't The following result is used in the text. Since no proof seems to be accessible one is supplied. A7. If f(6,t) e R is 2n-periodic in B and almost periodic in t then there exist (eo,t0) so that Vf(eo,t0) = (0,0) (V = gradient). EEQQI: If Vf(9.t) never vanishes let f(§,0) = . max f(e,0) and let (u(s),1(s)) define the steepest ascent SEED 921T] curve originating at (6,0). That is Eds- (v(s). r(sn -- valor) (4(0). 1(0)) = (5.0) u(s) and 1(5) exist for all s E (draw) since Vf(w,1) is bounded. Note that lim 1(s) = m for if |1(s)| s M < w define g(s) = s++w f(w(s), 1(5)) and set n = miSIVf(w(s), 1(5))[2 > 0, then for 52 s 2 0, g'(s) = lVf(p(s), 1(s))|2.3 n so that g(s) is unbounded which is absurd. Let f(p(1), 1(1)) - f(é,0) = a, a > O ~by assumption. Let T be an almost period in [1(1),1(1) + £(a/2)] relative to e = a/2 and s 2 1 so that 1(5) = T then 73 0/2 > f(w(S).1(S)) --f(v(5).0) 2 a + f(B,0) - f(w(5).0) ' = a + f(6,0) - f(w(s) - 21k,0) 2a. BIBLIOGRAPHY [I] [2] [3] [4] [5] [6] [7] [8] [9] [10] [ll] [12] [13] I BIBLIOGRAPHY Bogoliubov, N.N., and Mitropolsky, Y.A. (1961). Asymptotic Methods infithe Theorygof Non-Linear Oscillations. Hindustan Publishing Corp., Delhi. Bohr, H. (1947). Almost Periodic Functions. 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