A Vanderbauwhede Institute for Theoretical Mechanics, State University of Ghent
Local bifurcation and symmetry
Pitman Advanced Publishing Program BOSTON· LONDON· MELBOURNE
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© A Vanderbauwhede 1982 AMS Subject Classifications: (main) 58E07, 47H15 (subsidiary) 34C25, 35B32, 73H05 Library of Congress Cataloging in Publication Data Vanderbauwhede, A. Local bifurcation and symmetry. (Research notes in mathematics; 75) "Grew out of a Habilitation thesis ... presented at the State University of Gent in 1980"-Introd. Bibliography: p. Includes index. 1. Differential equations, Nonlinear-Numerical solutions. 2. Differential equations, Partial-Numerical solutions. 3. Bifurcation theory. I. Title. II. Series. 515.3'5 82-13249 QA372.V34 1982 ISBN 0-273-08569-7 British Library Cataloguing in Publication Data Vanderbauwhede, A. Local bifurcation and symmetry.-(Research notes in mathematics; 75) 1. Bifurcation theory I. Title II. Series 515.3'5 QA371 ISBN 0-273-08569-7 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording and/or otherwise without the prior written permission of the publishers. This book may not be lent, resold, hired out or otherwise disposed of by way of trade in any form of binding or cover other than that in which it is published, without the prior consent of the publishers. This book is sold subject to the Standard Conditions of Sale of Net Books and may not be resold in the UK below the net price. ISBN 0 273 08569 7 Reproduced and printed by photolithography in Great Britain by BiddIes Ltd, Guildford
To ANITA, BERT and tEEN
Preface
06 a Habili;ta.:LLon :the.-6--G6 C-Ln Vu;tch) wh1.ch WCL6 pILe.-6e-nte-d at :the. S:tate- Un-Lve.Mdy 06 Ge-nt in 1980. An Engwh ve-Mion 06 :th--U, :the.l>--G6 hCL6 cifLculate.d amo ng a f.,mail. gILoup 06 mathe.nlatiUanJ., wOlLizing in di6 ne-ILe-n;tial e.quationJ., and bi6UfLcarvCon :the.oILY. I:t ~~ :thILough :the. e.ncoulLageme-nt 06 f.,e-V(OfLal
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:the.-6--G6, and whof., e- patie-nce., inte-ILe.-6:t and pILa~cal adv--G6 e. have- be-(!.Yl a co nJ.,;tant f.,OuILce. 06 e.ncoulLag e.me-nt ciuJUng :the- PCL6:t ye.aM. On no le.-6f., impofLtance- We.ILe- my ILe.gulafL contaw wdh PIL06e.-6f.,OIL J. Mawhin 06 :the. InJ.,;t.J;tu;t de- Math~matique- in LouvcUn-la-Nwvc; I am parvt:iculafLly gfLate-6ul 60IL IUJ., 61Lie-ndoh-Lp, IUJ., f.,:timulaung e-nthuf.,iCL6m and moM 06 aU 601L IUJ., gILe-at mathe.matical f., IziU wh-Lch I could wdne.M ciuJUng :the- Jl!Ume-fLOU'-> f., e.minaM he. oILgan-Lze.-6. My pe-Monal inte.ILe.-6:t in bi6uILcation pILoblcmf., f.,tafLte.d dulLing :the. acade.mic ye.afL
1976-1977,
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06 BILown Un-Lve-f[f.,dy, PILovide-nce-. I am e.-6pe-uaUq gfLCc.:te.6ul :to PIL06e.-6MIL J.K. Hale- who 066c/'Le.d me- :t1UJ., un-Lque- oppofLtundy, 601L IUJ., manq advice.o and 601L h--U, 6~.e-ncL6h-Lp. H-Lo woILiz on bi6UfLcation :the.oILY hCL6 be-e-n v(OfLy f.,timulating nOlL my own ILe.-6e.afLch, and WCL6 an e-ve.Jr pILe.-6 e-nt guide- w~e- WfLiling :th--U, booiz. I aloo owe- much :to aU :thof.,e. who pILovide-d me- dulLing my f.,;tay at BILown Un-Lve.Mdy wilh a Mud bCL6--G6 nOlL my nufLthe.IL wOlLiz : J. MaUe.:t-PafLe.:t, who intfLoduce.d me- :to bi6llILCatiO n :the-OILY, V. He.My, T. BanQJ." F. Kapp~, J. I n6ante-, W. S;tf[aUJ.,f., and C. Va6e.fLmof., 60IL :thw le-ctUfLe.-6 and f.,e.minaM on di6ne-ILe.ntial e.quationJ." P. Tabow., 60IL many cLWcU'->f.,ion6, and H. M. RocLUgue.-6 nOlL the. pILe.tiy coUabolLation. Finally I aciznowle-dge- :the. Sue-ntiMc Commil;te-e. 06 NATO 60IL pILoviding :the. Mnanual wppofLt bOIL :th--U, f.,;tay.
1 "''''
w thank
ha, ee,
N. C
O. V.iehmann, S."N. Chow, V. Ch.i.U.i.ngwoWi and
1
M. Gofubillky 60IL a vwmbe!L 06 cL0.sCIL6J.JioV1.f., which J.J.{wr.:ted pcur.:t 06 :the wOfLk can-I 'i .to.J..ned in :thilJ book 0fL which in6fuenced :the pILuC',ntilion. I am aUo gfLa:te6u.t!, :to N. VanceIL 60fL painting aut an e!LMIL in :the eOJLUeIL dtLatt 06 :the book.
Finatty, I :thank PIL06uJ.J01L
K.
KifLchg~J.Jne!L and Pitman Pub!i6hing 60IL
:the
inte!Lu:t which :they :took in :the pubucilion 06 :thi6 book; my coUeaguu at -the I V1.f.,;tU:ute 06 TheolLeticai MechaniC-6, 60fL :thw mOfLai J.JUppOfL:t; my 6amUy 60fL endufLing :the J.Jide - e6 6ec:t6; V. ROM 60fL :taking calLe a 6 :the 6-4J UfLe6; and
H. VefLmi6 60IL :the ex:tfLeme calLe wLth which he pILepalLed :the mal'lMcfLipL
A. Vande!Lbauwhede, Gent, May 198'2.
I
Contents
Q-lAPTER 1. INTROrucrION rnAPTER 2. MA1HEMATICAL PRELIMINARIES
2.1. Some results from functional analysis 2.2. Periodic solutions of periodic linear ordinary differential equations 2.3. Elliptic partial differential equations 2.4. The von Karmful equations 2.5. Group representations and equivariant projections 2.6. Irreducible group representations rnAPTER 3. SYMMETRY AND 1HE LIAPUNOV-SrnMIDT ME1HOD
3.1. 3.2. 3.3. 3.4. 3.5. 3.6.
The Liapunov-Schmidt method Equivariant mappings The Liapunov-Schmidt method for equivariant equations Symmetric solutions Application: reversible systems Further results and applications
CHAPTER 4. PERTURBATIONS OF SYMMETRIC NONLINEAR EQUATIONS
4.1. 4.2. 4.3. 4.4.
Introduction The abstract results Perturbations of equations with O(2)-symmetry Axisymmetric perturbations of a problem with O(3)-symmetry
12 12 22 30 46 63 69 88 88
93 96 99 102 111 125 125 126 135 147
rnAP1ER 5. GENERIC BIFURCATION AND SYMMETRY 5.1. 5.2. 5.3. 5.4. 5.5. 5.6.
Introduction Bifurcation problems with a quadratic dominant term Bifurcation problems with a cubic dominant term Generic and non-generic bifurcation Generic conditions and symmetry Application : the von Karman equations for a rectangular plate
rnAP1ER 6. SYMMETRY AND BIFURCATION AT MULTIPLE EIGENVALUES 6.1. 6.2. 6.3. 6.4. 6.5. 6.6.
Introduction The abstract results Symmetry and bifurcation from a simple eigenvalue Application: equations with O(2)-symmetry Bifurcation of subharmonic solutions A bifurcation problem with O(3)-symmetry
152 152 153 160 171 180 185 203 203 204 218 222 237 251
rnAP1ER 7. BIFURCATION PROBLEMS WIlli SO(2)-SYMMETRY AND
HOPF BIFURCATION 7.1. 7.2. 7.3. 7.4. 7.5.
Introduction Bifurcation problems with SO(2)-symmetry Generic bifurcation under SO(2)-symmetry Hopf bifurcation Degenerate Hopf bifurcation
G-IAP1ER 8. SYMMETRY AND BIFURCATION NEAR FAMILIES OF SOLUTIONS 8. 1. 8.2. 8.3. 8.4. 8.5.
Introduction Reduction to a finite-dimensional problem Symmetric solutions Application : periodic perturbations of conservative systems The bifurcation set : an example
REFERENCES INDEX
261 261 263 273 281 300 307 307 308 315 320 325 333 349
1 Introduction
Many problems in applied mathematics reduce, after the introduction of an appropriate model, to that of solving a set of equations, which in an abstract form can be written as M(x, A)
o.
(1)
In this equation x is the unknown of the problem, A stands for the relevant parameters of the model, and M is a mapping which is in general nonlinear. The basic problem of bifurcation theory is to study how the solutions of (1) change as the parameter A is changed. In order to make (1) a mathematically well posed and treatable problem we have to give precise definitions of the spaces X and A to which x and A belong, and of the mapping M. The elements of X will usually be functions of some time and (or) space variables, which have to satisfy certain smoothness, initial value, boundary value or range conditions. Taking these side-conditions into the definition of X and introducing.an appropriate topology will generally result in giving X the structure of a manifold, usually an infinitedimensional submanifold of some function space. Also the parameter A will belong to a manifold A; for practical applications it is usually sufficient to consider finite-dimensional A, but since some mathematically interesting problems require an infinite-dimensional parameter space, we will not impose any restrictions on the dimension of A. Finally, the mapping Min (1) is defined for (X,A) E X xA, and takes its values in a space (or manifold) Z, which in many cases will be an ambient space of X. The ultimate goal of the theory is-of course to give a description, as complete as possible, of the solution set of (1); i.e. of the set S
=
{(X,A)EXxA iM(x,A)
=o}
(2)
More precisely, since we have made the distinction between the unknown x and the parameter A. we are in fact interested in the A-sections of S :
SA = {XEX
I (X,A)ES}
(3)
The problem of bifurcation theory is to study how SA varies with A. The structure of the solution set can be rather wild and chaotic, as follows from the fact that for each closed subset S of Xx fI. one can find a continuous M such that the solution set of M coincides with the given subset S. This remark shows that we will have to restrict the class of mappings M in order to corne to an efficient treatment. Usually we will impose on M certain smoothness conditions. Another restriction which we impose onto ourselves is that we will make only a local study of S and SA' i.e. in a neighbourhood of a given point (XO,A O) E S. This is what we mean by "local bifurcation theory". The restriction to a local study has a few important consequences. First, we can use local charts in the manifolds X, Z and fl., and consequently we may assume that X, Z and fI. are Banach spaces. Second, it will appear that the linearization of M at the point (XO,A O) around which we work will play an important role in the theol~. Before we give some definitions, let us look at a simple but instnlctive example, namely the eigenvalue problem for a linear operator A E LORn). We take X = Rn , fI. = R and define M : TIP xlR -+ Rn by M(X,A) = Ax-AX. We know from elementary algebra that for most values of A the equation M(X,A) = 0 has only the zero solution. It is only when A equals a real eigenvalue of A that there will be a nontrivial subspace of solutions; at such eigenvalues there is a branch of nontrivial solutions bifurcating from the line {a} xR eRn xlR of zero solutions. This example shows that the most interesting parameter values are those at which the structure of SA changes; such critical parameter values will be called bi6unc~on point6 (we give a precise definition further on). In the . linear example above the bifurcation points are the real eigenvalues of A; in general bifurcation points are parameter values at which certain solutions appear, disappear, coalesce or split up into several branches. In the literature one can find several definitions of a bifurcation point. Classically (see e.g. Krasnoselskii [134], Crandall and Rabinowitz [50], Sattinger [193]) one considers a one-parameter family of equations (i.e. fI.=lR), and assumes that for each A ElR a solution X(A) of (1) is given, where X(A) depends continuously on A. Then bifurcation is defined with respect ot this given curve of solutions. 2
Here we will use the more general definition given by Chow, Hale and Mallet-Paret ([ 39] ,[ 84]). The idea is to consider all equations near to a given equation in a certain sense, and to study all solutions of these perturbed equations near a given solution of the unperturbed equation. The precise framework in which we will work is as follows. Let X and Z be real Banach spaces, andn c X an open subset. For each integer r, let Cr(n;Z) denote the Banach space of all r-times continuously Frechet-differentiable mappings m : n 7 Z such that IImli r = sup IIm(x) I +11 Dm(x) I + ... +IID r m(x)1I} < ()()
xEn
(4)
Here D is differentiation, and 11.11 denotes the norm in the appropriate space. Usually we will assilllle r ;;;. 1. Definition. Let r EWand A c Cr(n;Z). Let
o.
Xo
E n and ~ E A be such that (5)
Then mO is a binWl.c.a:ti.oYl point a;t Xo wUh lteApe.d :to A i f for each neighbourhood U of (mO'xO) in Cr(n;Z) xX we can find (m,x 1) and (m,x 2), both belonging to' un (Ax n) and with x1 f x 2 ' such that m(x 1) = m(x 2) = O. Roughly speaking, mO is a bifurcation point at Xo when there is nonuniqueness of the solutions of m(x) = 0 near xo' for some mEA near mO. With this definition in mind one can pose the following problems. Problem 1. Given A c Cr(n;Z) and (mO'xO) E Axn, satisfying (5), show that mO is a bifurcation point at Xo with respect to A. Problem 2. Given A c Cr(n;Z), (mO'xO) E Axn satisfying (5) and a neighbourhood U of (mO'x O) in Cr(n;Z) xX, describe the local solution set ((m,x) EU I mE A, m(x) = A}. It is clear that when we can solve problem 2, then also problem 1 can be solved. In this book we will mainly be interested in problem 2; in particular we will want to find the nwnber of solutions of m(x) = 0 with (m,x) E U, for each mEA close to mO. When A is an open subset of Cr(n;Z) we call this pro-
3
blem the ge.ftvUc. bi6UJLc.a;t[oft ptwblem; in the opposite case we speak about a Jc.u:tJUcte.d bi6UJLc.a;t[oft pJc.oblem (see [ 84]) . As an example of a restricted bifurcation problem, let w be an open subset of a Banach space A, n : w + Cr(Q;Z) a continuous mapping, and A = {n(A) I AEW}. Then the corresponding restricted bifurcation problem amounts to the study of SA for A E w, where SA is defined by (3), with M : Q x w + Z defined by M(X,A) = n(A)(X). A point AO E w will be a bifurcation point for (1) when there is an Xo E SAO such that for each neighbourhood U of (XO,A O) in X x A we can find (xl,A) E U and (XZ,A) E U such that xl F Xz and M(x 1,A) = M(XZ,A) = O. In case one has given a solution branch of (1), i.e. a continuous map w + Q such that M(X(A) ,A) = 0 for all A E w, then AO E w will be a bifurcation point at Xo = X(A O) if for each neighbourhood U of (XO,A O) in XxA one can find (X,A) E U with x F X(A) and M(X,A) = O. By a simple translation this can be reduced to the special case where X(A) = 0, VA E w; then we speak about "bifurcation from the trivial solution". All restricted bifurcation problems considered in this book will be of the type described above, and will be written down in the fonn (1). An important remark that can be made at this point is that by taking A = Cr(Q;Z) and by defining M(x,m) = m(x) , Vx E Q, Vm E Cr(Q;Z), the bifurcation problem for (1) coincides with the generic bifurcation problem. Therefore our fonnalism allows us to handle both generic and restricted problems. Now suppose that M(X,A) in (1) is continuously differentiable in the variable x. Let (XO,A O) E X x A be such that M(XO,A O) = 0 while DxM(xO,AO) E L(X,Z) is an isomorphism. Then it is an immediate consequence of the implicit function theorem (see section Z.l) that in a neighbourhood VxW of (XO,A O) in X x A equation (1) will have a unique branch of solutions of the fonn {(x* (A), A) I AE W}. This implies that AO is not a bifurcation point at xO. For this reason we will restrict our attention to neighbourhoods of such solutions (XO,A O) of (1) at which DxM(xO,A O) is not an isomorphism. In fact we will ask more about L = DxM(xO,A O) than just that it is not an isomorphism. One of the main hypotheses used throughout the book will be that L is a Fredholm operator; this means that it has a finite-dimensional kernel, while its range is closed and has a finite codimension. This hypothesis will allow us to apply the so-called Liapunov-Schrnidt reduction method. This method is based on a splitting of the spaces X and Z, and uses the implicit function theorem to solve part of the equations. The outcome of the method is
x:
4
that there exists a one-to-one correspondance between the solutions of (1) near (XO,A O)' and the solutions near (O,A O) of a similar equation: F(u,A)
o.
(6)
Here F is a mapping from ker L x A into a complement of the range of L in Z, F(O,A O) = 0 and DuF(O,A O) = O. This means that for each A near AO (6) forms .a finite set of scalar equations in a finite number of unknowns (the compOnents of u). So the Liapunov-Schmidt method, when it can be applied, reduces the infinite-dimensional problem (1) to the finite-dimensional problem (6). Several methods have been used to study the bifurcation equation (6) : the Newton polygon, rescaling techniques, the implicit function theorem, or combinations of those. Also topological methods, such as the topological degree or the Liusternik-Schnirelman method can sometimes successfully be used to prove existence of bifurcating solutions. Our main tool in this book will be the implicit function theorem, sometimes in combination with a rescaling of certain variables; we obtain our results by methods which are "elementary and require only the calculus, the implicit function theorem, and a small amount of geometric intuition" (Hale [87]). To get some feeling for the problems, let us look at a few very simple examples (Hale [ 80]), for which the bifurcation equation (6) coincides with the original equation (1). We let X = Z = A =R, and M(X,A) = X2-A. The solution set S and the number of solutions as depending on the parameter A are shown in the following figures :
x
x=o
Fig. 1
I
(0)
(1)
(2)
It is clear that A = 0 is a bifurcation point. If we let M(X,A) = X(X 2-A),
then A = 0 remains the only bifurcation point, and the pictures look as follows :
x
A=O
Fig. 2
I
(1)
(1)
(3)
As a third example, let X = Z = JR, Ii. = JR2 and M(X,A) = x3-A1x-Az- A simple
analysis of the graph of the function x * M(X,A) shows that the equation M(X,A) = 0 has' exactly one solution when 27A~ > 4A~' and three distinct solutions when 27A~ < 4A~. The set of bifurcation points is given by {(A 1 ,A 2)9R2 \ 2n~ =4A~}' and is represented by the cusp in Fig. 3. These bifurcation points correspond to those values of the parameters for which the equation has a double or a triple solution. Restricting attention to parameters values A = (A 1,0), we obtain again the picture of Fig. 2. If we keep A2 fixed at a value different from zero (say A2 > 0); then the solution set looks 'as in Fig. 4.
"2
X
(1)
(3)
"1
0
Fig. 3 6
C Fig. 4
"1
Let us now return to a generic bifurcation problem, of the form M(x,m) = m(x) = 0, where mE Cr(rl;Z) (r sufficiently large) and x E rlcX. Let (mO'xO) E Cr(rl;Z) xrl be such that L = DmO(xO) is a Fredholm operator, with dim ker L '" codim R(L) = 1. Then the bifurcation equation (6) takes the form F(u,m) = 0, where for each m near ~, F(u,m) is a scalar function of the scalar variable u. From the general properties of bifurcation functions it follows that k
FCu,mO) '" cu
+
k
O(lul )
as
lui
-+
0 ,
(7)
for some k > 2, and with c f O. In a later chapter we will prove that if k '" 2 in (7), then the bifurcation set is a submanifold of Cr(rl;Z), passing through mO and having codimension one. For m at one side of the manifold the equation mCx) = 0 has no solutions near xo' for m at the other side there are two solutions; on the manifold itself there is a double solution. When we consider a one-parameter restricted problem, with n : IR -+ Cr(rl;Z) such that nCO) = ma and such that n is transversal to the manifold of bifurcation points, then the restriction of the generic problem to the one-dimensional path described by n gives us precisely a picture as in Fig. 1. If k = 3 in (7), then the bifurcation set is formed by two submanifolds of codimension 1, forming a cusp along their common boundary, which is itself a submanifold of codimension 2. Inside the cusp there are 3 solutions, outside the cusp there is only one solution. When we consider a two-dimensional restricted problem such that the corresponding n is transversal to this generic bifurcation set, then we obtain a bifurcation picture as in Fig. 3. The bifurcation picture of Fig. 2 forms a non-transversal restriction of such generic bifurcation problem. In general we will call a restricted bifurcation problem which is transversal to the corresponding generic bifurcation set a ge.rwfUc Jtu;tJUde.d b'{6U!LcruoYl pJtoblem. For such restricted problem the bifurcation set will have the same features as the corresponding generic bifurcation set. For a restricted problem to be generic certain transversality conditions have to be satisfied; we will refer to these conditions as ge.ne.fUc condit,{on6. They will be generically (i.e. almost always) satisfied if the number of parameters of the restriction is large enough. Readers who have some acquaintance with singularity theory (see e.g. Golubitsky and Guillemin [72], Brocker [25-26], Martinet [261], Poston and 7
Stewart [ 176]) will recognize in the foregoing discussion a number bf which appear also in this singularity theory. Indeed, the basic ideas of generic bifurcation theory were very much influenced by the approach of ~~j.l~Ll-4 larity theory, and in recent years there has been an increasing interference between both theories (see e.g. Marsden [155]). Singularity theory was even at the basis of a new area of research in bifurcation theory, which started with the basic papers of Golubitsky and Schaeffer [74-75]; in this work singularity theory is used to classify perturbations of given bifurcation problems. In this book we will not use the methods of singularity theory, although it is undeniable that some ideas from this theory have influenced our presentation. Now let us come to the main subject of this book, which is the influence of the symmetry properties of the mapping M on its bifurcation properties. In many practical applications one obtains equations which exhibit a certain degree of symmetry, which is due either to some simplifications used in the model, or to some basic symmetry of the problem at hand. Mathematically, this symmetry will be expressed by means of commutation relations between M and a group of symmetry operators; this will give rise to a class of so-called equivariant operators. It is our goal to develop a systematic approach to bifurcation problems for such equivariant operators. It is clear that the equivariance of M will be reflected into a corresponding symmetry of the solution set S. Indeed, S will be a union of orbits of solutions, each orbit being generated from any of its members by application r of the symmetry operators. This forces us to reformulate our basic bifurcation problem, and to consider bifurcation near orbits of solutions. This modified problem will reduce to the classical bifurcation problem if the orbit reduces to one single point, i.e. if we work near a solution which is invariant under the symmetry operators.But then also the linearization at such solution will have the full symmetry of the problem, which in general results in a higher dimensional solution space for this linearized problem. Via the Liapunov-Schmidt method this will result in a higher-dilnensional, and therefore more difficult, set of bifurcation equations (6). Fortunately, if the Liapunov-Schmidt reduction is performed in an appropriate way, then also the bifurcation function F(u,A) will be equivariant, and this may compensate to some extent the difficulties arising from the higher dimensions. Group representation theory will play an important role in the analysis of these 8
equivariant bifurcation equations. Another remark is that equivariant bifurcation problem are in general nongeneric, in the sense that small perturbations will destroy the symmetry. Some (e.g. Sattinger [196]) even go a step further and suggest that the nongenericity of equivariant equations is at the origin of bifurcation for general systems : from this point of view, bifurcation is "closely related to symmetry breaking and bifurcation points necessarily have a degree of symmetry. C... ) Although systems with symmetries are nongeneric, they nevertheless play pivotal roles as bifurcation points" (Marsden [ 155]) . In the literature one can find many examples of bifurcation results whose proof relies on some symmetry of the equations; many times this fact is somewhat obscured by an ad hoc formulation of the symmetry properties. It is only by a systematic approach that the underlying structures induced by the symmetry can be revealed. The development of such systematic approach within the framework of modern bifurcation theory has been our guiding motive for writing this book. We hope that our approach will also be useful when dealing with other aspects of bifurcation theory which are not treated here. SURVEY OF THE mNTENTS Chapt~
2 contains most of the technical material needed further in the book.
First we review some results from functional analysis, in particular Fredholm operators and the implicit function theorem. Then we give a number of results on three different particular problems which will be used throughout the book to illustrate our abstract results; these three problems are (i) the determination of periodic solutions of periodic or autonomous differential equations (section 2.2), (ii) some elliptic boundary value problems with Dirichlet data (section 2.3) and, (iii) the buckling problem for plates subject to thrust and normal load, as described by the von Karman equations (section 2.4). In section 2.5 we define group representations over general Banach spaces, and describe a few results on associated projection operators. Section 2.6 contains some basic results on irreducible representations, and describes the irreducible representations of the groups 0(2) and 0(3). In ehapt~ 3 we introduce the Liapunov-Schmidt fine equivariant mappings (section 3.2) and study symmetry and the Liapunov-Schmidt method (section furcation equations reduce for symmetry-invariant
method (section 3.1), dethe interference between 3.3). We show how the bisolutions (section 3.4) 9
and apply the theory to reversible periodic systems (section 3.5). A careful analysis of this example reveals some further properties which can easily be generalized and applied to other problems (section 3.6). In ehapt~ 4 we obtain some conditions which ensure that all bifurcating solutions of small perturbations of an equivariant equation will have the symmetry of the perturbation. Such result was proved by Hale and Rodrigues for the Duffing equation [88], and later generalized by Rodrigues and the author [182]. A further generalization is given in section 4.2, while the sections 4.3 and 4.4 contain applications to perturbations of problems with respectively 0(2) and 0(3) symmetry. Chapt~ 5 contains a discussion of generic bifurcation in the important case that the bifurcation equation (6) is a scalar equation in a scalar unknown u. As already mentionned in the introduction, the bifurcation behaviour depends highly on the value of k in (7); in section 5.2 we discuss the case k = 2, and in section 5.3 and 5.4 the case k = 3. In section 5.5 we show how symmetry leads to nongeneric bifurcation. As an application we study in section 5.6 how the symmetry of the normal load may affect the bifurcation behaviour for the buckling problem of a rectangular plate. In ehapt~ 6 we turn to the bifurcation problem at multiple eigenvalues for equivariant mappings. The abstract theory is given in section 6.2, and contains as a special case the classical theorem of Crandall and Rabinowitz [SO] for bifurcation from a simple eigenvalue (section 6.3). The remaining sections of chapter 6 contain applications to equations with 0(2)-symmetry , (section 6.4), to the bifurcation of subharmonic solutions (section 6.5) and to a boundary value problem with 0(3)-symmetry (section 6.6). The main application of the results of section 6.2 is discussed in ehapt~ 7, where we study bifurcation for problems with SO(2)-symmetry. An important example of such bifurcation problem is the so-called Hopf bifurcation, which describes the bifurcation of periodic solutions from stationary solutions of autonomous differential equations. Section 7.2 contains the general theory for problems with SO(2)-symmetry, section 7.3 discusses genericity under SO(2)-symmetry, and section 7.4 gives a unified treatment of Hopf bifurcation for both ordinary and functional differential equations. In section 7.5 we give some examples of degenerate Hopf bifurcation; one of the results is the Liapunov Center theorem, which follows here from our Hopf bifurcation analysis. 10
In the final ~hapte~ 8 we study bifurcation near a compact orbit of solutions of the unperturbed equation. In section 8.2 we reduce this problem to a finite-dimensional one, while in section 8.3 we show how the symmetry of the equations may sometimes imply the existence of particular symmetry-invariant solutions. In the final section 8.4 we discuss the bifurcation equations for a particular example, namely that of periodic perturbations of a conservative oscillation equation. We conclude this survey with a remark on the organisation of the references within the text. When we want to refer to a particular result, say theorem 5.2.5, then this will be done in one of the following ways: (i) theorem 5, when the reference appears in section 5.2; (ii) theorem 2.5, when the reference appears in another section of chapter 5; (iii) theorem 5.2.5, when the reference appears in another chapter. A similar system will be used for formulas : formula (7) means formula (7) in the same section, formula (3.7) refers to formula (7) of section 3 of the same chapter, and formula (4.3.7) refers to formula (7) of section 4.3. Bibliographical note. Since some time the literature on bifurcation theory is growing almost exponentially; therefore the next list of references just intends to give a key to this vaste field of research. There are a number of books, lecture notes and conference proceedings which should provide a good introduction to bifurcation theory and its applications; we mention in particular : Chow and Hale [245], Hale [80J,[ 252J, Sattinger [193J,[ 265], Pimbley [ 173], Iooss [ 101], Stuart [211], Keller and Antman [ 114], Berger [ 21 ], Krasnosel'skii [134], Ize [103], Rabinowitz [164], Amann, Bazley and Kirchgassner [241] and Iooss and Joseph [255J. Further we want to mention the papers by Crandall and Rabinowitz [50J, Dancer [55J, Sather [187], Stakgold [209], Westreich [235], Chow, Hale and Mallet-Paret [39J, Golubitsky and Schaeffer [74], Rabinowitz [180J, Keener and Keller [111], Marsden [155] and Sattinger [266].
11
2 Mathematical preliminaries
It is the aim of this chapter to provide the mathematical background needed for the treatment in the subsequent chapters. We start with some results from functional analysis, such as the Fredholm alternative for compact operators, the Lax-Milgram theorem, the Krein-Rutman theorem, the implicit function theorem and the rank theorem. In sections 2, 3 and 4 we describe the three basic examples which will be used in later chapters to illustrate the abstract results. These are the determination of periodic solutions of periodic ordinary differeIltial equations (section 2), elliptic boundary value problems on bounded domains in Rn (section 3), and the. von Karman equations which describe the buckling problem from plate theory (section 4). Most of the results are stated without proofs; however, we have included a number of intermediate results, which may help to give an idea of how the final conclusions can be obtained. We also give ample bibliographical information on more complete treatments. Sections 5 and 6 give an elementary introduction to group representation theory. In section 5 we introduce the concept of a representation of a group over a linear space, and prove the existence of an equivariant projection on an invariant subspace when the group is compact. In section 6 we prove some " results on irreducible representations, and we discuss the examples of the 3 - m 2 and m rotatlon groups ln~ ~ . As a general remark, which holds throughout this book, let us mention that we consider only ~~a{ spaces, except when the contrary is explicitly stated. 2.1. SOME RESULTS FROM FUNCTIONAL ANALYSIS 2.1.1. Definition. Let X be a topological vector space, and M a closed subspace. Then M has a topolog~~a{ ~omplem~nt ~n X if there exists a closed subspace N of X, which is an algebraic complement of M :
12
x=
Mn N
M + N
=
{O} ,
(1)
and such that the map (x 1 ,x 2) ++ x 1+x 2 from the topological product MxN onto X is a homeomorphism. We say that X is the eLUted ,-,urn of M and N, and we write: X=MEBN.
(2)
2.1.2. Theorem. A closed subspace M of a topological vector space X has a topological complement if and only if there exists a continuous projection P in X such that M
= R(P)
(3)
Then we have X = MEB N, with N to M. D
ker P, while X/N is topologically isomorphic
2.1.3. Theorem. Let ~ be a closed subspace of a topological vector space X. (i) If X is locally convex, and dim M < 00, then M has a topological complement in X. (ii) If dim(X/M) < 00, then M has a topological complement in X; each algebraic complement of M is also a topological complement. D 2.1.4. Theorem. Let M be a closed subspace of a Hilbert space X. Then M has a topological complement in X, and D
(4)
In the sequel subspaces with a topological complement will mostly be associated with so-called Fredholm operators. 2.1.5. Definition. Let X and Z be two Banach spaces, and L : X ~ Z a bounded linear operator. Then L is called a F~edhofm ope~ato~ if the following conditions are satisfied : (i) dim ker L < 00 ; (ii) R(L) is closed (iii) codirn R(L) < 00
•
13
The element of:Z given by ind L = dim ker L - codim R(L) is called the
~ndex
(5)
of the Fredholm operator L.
2.1.6. Theorem. Let X and Z be Banach spaces, and L E L(X,Z) a Fredhom operator. Then there exist continuous projections P E L(X) and Q E L(Z) such that R(P) = ker L
ker Q = R(L) .
(6)
Moreover, the restriction of L to ker P has a bounded inverse K ker P, satisfying
KLx
(I-P)x
PK(I-Q)z
o
LK(I-Q)z Yx E X
=
R(L)
~
(I-Q)z Yz E Z
(7)
Proof. The first part follows from the theorems 2 and 3. The restriction of L to ker P is a continuous linear bijection between the Banach spaces ker P and R(L). So it has a bounded inverse K, by the open mapping theorem (see Rudin [ 184] , theorem 2. 11 and corollary 2. 1 2) . D The result of this theorem will be the main tool used in the LiapunovSchmidt reduction method given in chapter 3. A particular class of Fredholm operators is given by the compact perturbations of the identity. 2.1.7. Definition. Let X and Z be Banach spaces. An operator L E L(X,Z) is called ~ompact if L transforms bounded subsets of X into relatively compact subsets of Z. 2.1.8. Lemma. Let L E L(X) be compact. Then I-L is a Fredholm operator with zero index. 0 2.1.9. Lemma. Let L E L(X,Z). Then L is compact if and only if L* E L(Z* ,X*) is compact. If X = Z, then also
14
dim kerO-L)
o
dim kerO-L*)
r
2.1.10. Theorem. Let L E L(X) be compact, and A O. Then either the linear operator U-L has a bounded inverse, or A is an eigenvalue of L with finite multiplicity. The set of eigenvalues of L is at most countable, and can only have 0 as an accumulation point. Further, A f 0 is an eigenvalue of L if and only if A is an eigenvalue of L We also have : (kerCU-L*))
R(U-L)
1
(8)
1 .
(9)
and RCU-L*)
=
Cker(U-L))
Finally, if A f 0 is an eigenvalue of L, then there exists an integer k such that X = ker(U-L) k ffi RC',I-L) k
o
2.1.11. Remark. Theorem 2.1.10 contains the so-called F~edhotm for compact operators. This can be formulated as follows Let L E L (X) be compact. Then either the equation
~
1
(10) att~native
x-Lx = Y
(11 )
has a unique solution x for each y E X, or the homogeneous equation x-Lx
0
(12)
has a nontrivial solution. In the first case the operator (I-L)-l is bounded, in the second case the equation (11) has a solution if and only if : <x* ,y>
o
(13)
15
for each solution x* E X* of the homogeneous adjoint equation X'-L*X'
o.
C14)
Now we state two basic theorems on representations of bounded linear functionals over Hilbert spaces. 2.1.12. Riesz representation theorem. For every bounded linear functional F on a Hilbert space H, there is a uniquely determined element f E H such that FCx) = (x,f) for all x E H; also ~F~ = ~f~. [] 2.1.13. 'The Lax-Milgram theorem. Let B : HxH bilinear form over a Hilbert space H, i.e. :
-+
JR be a bounded
COe}LUVi?-
iB(x,y)j < KlxllyR ,
Vx,y E H
CIS)
Bex ,x) ;:;, vii x~ 2
Vx
(16 )
and E
H
for some K > 0 and v > O. Then for every bounded linear functional F there exists a unique element f E H such that F(x)
B(x,f)
Vx
E
H •
E H* ,
( 17)
Moreover [J
l18)
We will also need some results from the theory of ordered Banach spaces, more in particular the Krein-Rutman theorem. More details can be found in Krein & Rutman [136], KrasIlOselskii [133] , and Amann [5] . 2.1.14. Definitions. Let X be a real vector space. A subset Pc X is called a coni?- if the following is satisfied :
16
(i)
x+y E P
(ii)
AX E
Vx,y E P ;
P
VA E JR+ ' Vx E P
(iii) pn(-p) o Such a cone defines an X
ohd~~ng
in X, as follows
< y if and only if y-x E P .
P is called the pO-6itiv~ ccon~ of the ordering. _An ohd~~d Banacch -6pacc~ is a
Banach space X together with a closed positive cone P. We will use the notation
,P) and the shorthand OBS.
The cone P is g~n~a;t 0 ; rCA) is a simple eigenvalue of A, having a positive eigenvector; rCA) is also a simple eigenvalue of A*, having a strictly positive eigenvector;
(iv)
no other eigenvalues of A correspond to positive eigenvectors;
they all satisfy 1\ I < rCA) . We finish this section by stating a version of the implicit function theorem as we will use it further on. It forms the key step at many places in the further theory. The version quoted below is the one given by Rabinowitz in [ 180] . For the proof, one can adapt the arguments given in the proof of theorem 10.2.1 of Dieudonne [249]. 2.1.16. The implicit function theorem. Suppose (a) X, Y and Z are three (real) Banach spaces; (b) f is a continuous mapping of an open subset A of XxY into Z; (c) if we set Ax
=
{yEY : (x,y)EA}
then, for each x E X such that Ax f ¢, the map y * f(x,y) of Ax into Z is Frechet-differentiable in Ax' and the derivative of this map (denoted by Dyf) is continuous in A; Cd) (xo'YO) E A is such that f(xo'YO) = 0 and DyfCxO'YO) is a linear homeomorphism of Y onto Z. Then there exist neighbourhoods U of Xo in X and V of YO in Y, and a continuous mapping y* : U -+ V such that (i)
UxV c A ;
(ii) for each (x,y) E UxV we have f(x,y)
o if and only if
y = y" (x) .
If moreover the mapping f is k-times continuously differentiable on A, then also y* is k-times continuously differentiable on U. 0 Two results which are closely related to the implicit function theorem are the inverse function theorem and the rank theorem. We
~tate
the inverse
nlnction theorem, and prove an infinite-dimensional version of the rank theorem.
18
2.1.17. The inverse function theorem. Let X and Y be Banach spaces, Ii c X open, Xo E [2, and f : Ii -7- Y a mapping of class C1 • Then there exists an open ne ighbourhood U of Xo (wi th Ueli) such that f (U) is open and f is a C1_ diffeomorphism of U onto f(U) , if and only if Df(x O) is a linear homeomorphism of X onto Y. 0 2.1.18. The rank theorem. Let X and Y be Banach spaces, Ii C X open, Xo E Ii and f : Ii -7- Y a mapping of class C1 • Suppose that there exist closed subspaces Xl and Y1 of respectively X and Y, such that: (i)
X = ker Df(xO) @ Xl ;
(ii)
R(Df(x)) is closed in Y , Vx E Ii
(iii) Y
=
R(Df(x)) @Y 1 ,Vx En.
Then there exist Cl-diffeomorphisms n : Ueli defined respectively in a neighbourhood U of V of f(x O) in Y, such that n(x O) = 0, Dn(x O) Dc;(f(x O)) = I y ' feU) c V and Vx
E
n (U) C X and c; : V C Y -7- C; (V)cY, Xo in X and in a neighbourhood = IX' c;(f(x O)) = 0,
-7-
n(U)
(19)
°
Proof. We may suppose that Xo = and f(O) = O. Let L = Df(O), and let P E LeX) and Q E LCY) be projections such that Rep) = ker L, ker P = Xl' ker Q = R(L) and R(Q) = Y1 (see theorem 2). By the same argument as in the proof of theorem 6 the restriction of L to Xl has a bounded inverse K : R(L) -7- Xl' satisfying relations similar to (7). Define n : Ii -7- X by Px
nCx)
+
K(I -Q) f(x)
Vx En.
(20)
Since nCO) = 0 and Dn(O) = P + K(I-Q)L = IX' the restriction of n to a suitahIe neighbourhood U of 0 in n is a C1-diffeomorphism. We may suppose that n(U) is convex and such that x E n(U) implies (I-P)x E n(U). Let f1 = fon-1 : neU) -7- Y. Applying L on the identity x
=
Pn
-1
ex)
+
K(I-Q)f 1 ex)
Vx
E
n(U)
(21)
19
we find (I-Q)f 1 ex) = Lx
Vx
E
n(U) .
(22)
It follows that Df 1 (x).Ph E Y1 for x E nCU) and hEX. Since by hypothesis R(Df 1 (x))nY 1 = {O}, we conclude that Df 1 (x)Ph = 0 for x E n(U) and hEX. Then the mean value theorem gives f l (x) = f 1((I-P)x)
Vx
E
n(U) .
Let V = {yEYiK(I-Q)yEn(D)} and define s : V.-r Y by s(Y) = Y - Qf 1 (KCI-Q)y)
Vy
(23)
E V •
Since K(I-Q)s(Y) = K(I-Q)y it is straightforward to show that s is a C1_ diffeomorphism, with s(V) = V and s-l V .-r V given by s
-1
(y) = y
+
Qf 1 (K(I-Q)y) ,
Vy
E
V .
From (22) we obtain K(I-Q)f 1 Cx) CI-P)x E n(D) for x E nCD), so that fCD) = f1 (n(D)) C V. Then (19) follows from a direct calculation.
0
2.1.19. Corollary. Suppose that f : r2 C X .-r Y is of class C 1 and such that
Rk fex) = dim R(Df(x)) = k
for some kEN. Then at each point Xo E theorem holds.
Vx r2
E r2 ,
(24)
the conclusion of the rank
Proof. We show that the conditions of the rank theorem are satisfied in a neighbourhood Si' C n of xo. The existence of X1 follows from the fact that ker Df(x O) has a finite codimension, by (24). Also condition (ii) follows trivially from (24). In order to show (iii), remark that there exists a closed subspace Y1 of Y such that (iii) is satisfied for x = xo. From (24) we have codim Y1 = k, and it is sufficient to show that R(Df(x)) nY1 = {O} for all x near xo. Let P and Q be as in the proof of the rank theorem. Then (I-Q)Df(xO) is an
isomorphism between Xl and R(Df(x O))' Consequently (I-Q)Df(x) remains an isomorphism between Xl and R(Df(xO)) for all x near xO' So we have necessarily dim Df(x) (Xl) > k, and then the hypothesis implies that Df(x) (X 1) = Df(x)(X). If now y E Df(x)(X)ny l , theny = Df(x)h for some h E Xl' Since y E Y1 it follows that (I-Q)Df(x)h = O. Because (I~Q)Df(x) is an isomorphism between Xl and R(Df(x O)), we conclude that h = 0 and y = O. This proves the corollary.
0
2.1.20. Corollary. Under the conditions of theorem 18 or corollary 19 each point Xo E ~ has a neighbourhood U such that feU) is a C1-submanifold of Y. Also f- 1 (f(x O)) is a Cl-submanifold of ~ for each Xo E r/. Proof. This follows immediately from the conclusion of the rank theorem and the definition of a submanifold (see e.g. Lang [141]). 2.1.21.~ibliographical
0
notes. The results about topological complements are
taken from Kothe r 257] . The theory of Fredholm operators, and more in particular of compact operators, can be found in any textbook on functional analysis; one can see for exaTJlple Rudin [184] , Schechter [198], Lang [140J, Taylor [215] , Yosida [237] ; more detailed results can be found in Kato [ 108] and Dunford and Schwartz [ 59] . The Lax-Milgram theorem is the basic result used. in the modern Hilbert space approach to boundary value problems for elliptic partial differential equations: see e.g. Schechter [ 199J , Showalter [207] , Gilbarg and Trudinger [ 711 , Bers, John and Schechter [ 221 , etc ... Finally, the implicit function theorem (and the associated Banach fixed point theorem) fonn the start of almost any treatment on nonlinear functional analysis. A nice account can be found in Schwartz [ 203] . Since it is a local result, it is also frequently used in the theory of differentiable manifolds (see e.g. Lang [ 141] , Chillingworth [ 37] ). In the following sections and chapters it will be our unique tool for solving equations. A different fonnulation of the rank theorem can be found in Bourbaki [243], while Dieudonne[249] gives a finite-dimensional version of this theorem.
21
2.2. PERIODIC SOLUTIONS OF PERIODIC LINEAR ORDINARY DIFFERENTIAL EQUATIONS In this section we review the Fredholm alternative for the periodic solutions of periodic linear ordinary differential equations. 2.2.1. The set-up of the problem. Let us look for 2n-periodic solutions of the homogeneous linear equation x
A(t)x
=
(1)
and its non-homogeneous counterpart x = A(t)x + f(t)
(2)
Here A : lR -+ L ~1) and f : lR -+ lRn are 2n-periodic and continuous. We will also have the occasion to consider the (formal) adjoint equation
.
-y
T
A
=
(t)y .
(3)
Since our interest will be in 2n-periodic solutions, let us introduce the following Banach spaces :
z=
{x: lR-+lRn I x is continuous and 2n-periodic}
x=
{xE Z
and
I XE Z}
with the usual CO, respectively C1 topology IIzllZ
sup{l!z(t)lIltElR},
and IIXII X
= sup{\!x(t)ll+lIx(t)UltElR} .
With the equations (1) and (2) we associate the bounded linear operator 22
L
X -+ Z
x(.)
+*
(Lx)(.)
=
dx at(.) -A(.)x(.)
(4)
and its formal adjoint L* : X -+ Z , x(.)
ff
(L*x)(.) =-~~(.) -AT(.)x(.)
(5)
We want to show that L is a Fredholm operator, and give a description of ker L and R(L). In doing so we will use the following bilinear form over Z 2IT
<x,y> = f 0
VX,y E Z
(x(t) ,yet) )dt
(6)
where (.,.) is the usual scalar product inRn .
2.2.2. Some elementary results. The right-hand side of (1) and (2) is continuous in (t,x), and, by the linearity in x, also Lipschitz-continuous in x. As a consequence, the Cauchy problem formed by the equation (1) or (2), together with the initial condition x(t O) = x o ' has a urUque. solution for all (to'x O) E R xRn; this solution exists and is of class C1 for all t E R. This basic existence and uniqueness result has several consequences. First, if we denote by ¢(t,t O) the matrix solution of (1) satisfying the initial condition X(t O) = I (= the identity matrix),then ¢(t,t O) is nonsingular for all t,to E R, and we have the following relations: ¢(to,t o)
=
I
Vto E R , (7)
¢(t,t 1 )¢(t 1 ,t O)
=
¢(t,t O)
Vt,t O,t 1 ER .
¢ : RxR -+ LORn) is called the bWyiJ.,~t.ioVl ma;t/Ux for equation (1). Indeed, the solution of (1) satisfying the initial condition x(t O) = Xo is given by (8)
the analogous solution of (2) is given by x(t;to'xo )
=
¢(t,tO)x O +
It
¢(t,s)f(s)ds
(9)
to
this
lS
the so-called VaJUaXiOVl-On-c.olUmnt6nOJunuia. 23
The adjoint equation (3) has similar properties; its transition matrix is given by T W (to,t) .
(10)
2.2.3. Periodic solutions. Another consequence of the uniqueness result is the following criterion for periodicity of solutions : a Mfutiovt x(t) Ob (1) OIL (2) --L6 2'Tf-pe/uoiUc- ib and Ovt-Ly i6 x(O) = x(2'Tf). Using (8) thjs gives the following condition for the initial value of a periodic solution of (1) :
where C
= w(2'Tf,O)
ker L
is a
mOvto~omy-matnix
for (1). We conclude that
{x(.) =w(.,O)xO I xOEkerCI-C)}
(11)
Since the monodromy-matrix for the adjoint equation (3) is given by (12)
we also have ker L*
{y(.) =q,T(O,·)yo T
{y(.) =q, (O,·)yo
I yoEker(I-C*)} I yoEkerCI-CT )}
( 13)
It follows that
dim ker L = dim ker L*
~
n .
(14 )
For the non-homogeneous equation (2) the periodicity condition can be obtained using the variation-af-constants farnula. We find :
Xo
2'Tf
= CxO
+
J0 q,(2'Tf,s)f(s)ds.
The equation (15) has a solution 24
Xo
(15) ERn if and only if
2TI (YO' f 0 ¢(2TI,s)f(s)ds) = 0
VyO
E
T
(16)
ker(I-C ) T
Writing ¢(2TI,s) = ¢(2TI,O)¢(O,s) = C¢(O,s), and observing that C YO T YO E ker(I-C ), condition (16) is equivalent to :
(YO,f~TI¢(O,S)f(S)dS)
= 0
Yo for
T
Vy 0
E
ker (I -C )
VyO
E
kerCI-C )
or
r2TI (¢T (O,s)YO,f(s))ds
)0
= 0
T
or still, using the characterization (13) of ker L*
=
Vy
0
ker L*
E
(17)
So (17) is a necessary and sufficient condition for f to belong to R(L). From (14) we can then conclude that L is a Fredholm operator with zero index. 2.2.4. We can also give an explicit form for the projections PE L(X), QE L(Z) and for the operator K: R(L) + ker P given by theorem 1.6. Let dim ker L= dim ker L* = P < n, and let
1* . Since is a positive definite symmetric bilinear form over Z (and so also over X), we may assume that {¢i1i=1, ... ,p} and {~ili=l, ... ,p} form orthonormal sets with respect to this form :
be bases for ker L, respectively ker
= I
J
I
J
= 0- 1J
Vi,j
1 , ••• ,p .
(18)
Then we can define P and Q as follows P
Px
I
i=l
<x'¢i>¢i
Vx
E X
(19)
25
and p
Qz =
I
i=1
¥z E Z •
1/Ii
(20)
It is easily checked that P and Q satisfy the necessary conditions. The pseudo-inverse K of L takes the form : Kf = where
Xo
(I-P)[~(.,O)xO .
+
f· ~(.,s)f(s)ds]
(21)
0
= xO(f) is any solution of
21T (I-C)xO = f0
~(21T,s)f(s)ds.
(22)
2.2.5. Example. Consider the second order scalar equation x
+
x
=
f(t)
(23)
where f(t) is 21T-periodic and continuous. Taking Z as before, and X = {xE Z I XE Z, xE Z} we can associate with- (23) the linear operator L : X + Z, xC.) * (Lx)(.) = x(.) +x(.) .
(24)
Rewriting (23) as two scalar equations of first order, it is immediate to see that : ker L = ker L* = {aeos(.) +bsin(.)
I (a,b) E]Rh
•
(25)
(Indeed, the homogeneous counterpart of (23), x+x = 0, is formally adjoint ). We also have: R(L) = {fE Z
I
f:
1T f(s)eoss ds =
f:
1T f(s)sins ds = 0 }.
self-
(26)
In the definition of P and Q we can take = -
1
lIT 26
eos t
- w(h) and h ]h O-6,h O+6[, and w(h O)
=
x(h) are continuously differentiable in
k (43)
x(hO)(t) = x(hO)Ckt) for some k
+
( 42)
E
=
xO(t)
'It E R
N\{O}.
2.2.8. The foregoing gives us a two-parameter family of solutions of (28),
of the form : x(t;h,e) = x(h) (t+e)
=
x(h) (w(h) (t+e))
Derivation in the parameters gives us two linearly independent solutions of the variational equation, namely
and
If we assume that
~T (h ) f 0 dh
(44)
0
then y 1 (t) is non-periodic. We conclude that under the condition (44) ker L = {axO I aER} .
(45)
Rewriting the nonhomogeneous equation
x + g'(xoCt))x
= f(t)
(46)
29
as a system of two first order equations, and applying the preceding theory, one finds that
f
21T R(L) = {fE Z I 0 f(s)xO(s)ds = O}
(47)
Finally, we can take for the projections P and Q
and
fo21T f(s)xO(s)ds
Qz
Vz
E
Z
( 48)
where a
=
2 f021T xO(s)ds
.
( 49)
We will use these results in chapter 8. 2.2.9. Bibliographical notes. The main results of this section can be found in any of the numerous textbooks on ordinary differential equations, for example Hale [ 77 ], Coddington and Levinson [ 46 ], Rouche and Mawhin [ 183 ], Knobloch and Kappel [ 129], etc. 2.3. ELLIPTIC PARTIAL DIFFERENTIAL OPERATORS In this section we collect some results on boundary value problems for second order elliptic partial differential operators. In particular, we will pay attention to the Dirichlet boundary value problem for the Laplacian. MJre details and proofs can be found e.g. in Friedman [65],[ 66], Bel'S, John and Schechter [ 22 ], Gilbarg and Trudinger [ 71], Folland [ 64 ], Miranda [ 167], Agmon [3], Agmon, Douglis and Nirenberg [2J. We remark that all functions considered are real-valued. 2.3.1. Definition. Let r2 ERn be an open bOUYlde.d domain. (We assume n ;;;. 2). Consider the linear differential operator
30
n
(l2 aij(x)()x.(lx.
L i ,j=1
L
n
L
+
i=1
J
1
°-
where a .. ,b. ,c E c (Q), and a. . 1J 1 1J each point x E Q, we have : n
L
i,j=1
a .. (x)~.~. 1J 1 J
>
b. (x)~ 1
aX i
+
c(x)
(1)
a.·. We say that L is e£.Upuc. when, at J1
°
(2)
We say that L is -6;t;Uc;tty e£.Upuc. when there exists a constant AO > Osuch that: n
L
i,j=1
a .. (xn·~· 1J
1 J
;;.
Aol~1
2
'Ix En,
V~
E JRn •
(3)
In fact, under the conditions imposed on the coefficients of L, this implies that L is uMbo/unly e£.Upuc.. 2.3.2. Theorem (Weak maximum principle). Let L be elliptic, with c(x) 'Ix E Q. Suppose that u E C2 CQ) ncO (n), and that Lu(x) ;;.
°
'Ix E Q .
0,
(4)
Then the maximum of u in Q is achieved on (lQ, that is o
sup u(x) = sup u(x) . xEQ xE(lQ
(5)
°
2.3.3. Corollary. Let L be elliptic, and c(x) < for all x E Q. Suppose that u E C2 CQ) nCO(Q) and LuCx) ;;. 0, 'Ix E Q. Then +
sup u(x) < sup u (x) xEQ xE(lQ
(6)
where u + (x) = max(u(x),O). If Lu(x) = 0, 'Ix E Q, then sup lu(x) xEQ
I = sup
xE(lQ
lu(x)
I .
o
(7)
°
2.3.4. Theorem. Let L be elliptic, and c(x) < for x E Q. Let u and v belong to C2 (Il) ncO (n), and suppose Lu(x) = Lv (x) , 'Ix E Q, and u(x) = vex), 31
Then u(x) = vex) for all x E [I. 0 This theorem proves uniqueness for solutions of the c-ta.6,-,ic-ai Vvuc-htu pfLObtem for the operator L : given f E CO(r2) and cp E CO(ri) , find u E C2 (r2) n CO (ri) such that
Vx
E d[l.
Lu(x) = f(x)
Vx
E
u(x) = cp(x)
Vx
E dr2
r2 (8)
However, the theorem tells nothing about the existence of such solution (see further).
°
2.3.5. Theorem. Let L be uniformly elliptic and c(x) = for all x E r2. If u E C2 (r2) ~CO(ri) satisfies Lu(x) ~ for x E r2, aild if u achieves its maxiIlllm in a point of [I, then u(x) is constant in ?2. If c(x) ~ in r2, then u cannot achieve a non-negative maxiilllli~ in \1 unless it is constant. 0 This theorem is the '-'thong maximum p~n~{ptc of E. Hopf.
°
°
°
2.3.6. Theorem. Let L be strictly elliptic and c(x) ~ in ~. Suppose that 2 u E C (r2) ncO(ri) and Lu(x) '" f(x) for x E r2, where f is a given bounded function on r2. Then: sup lu(x)
1
~
xE\1
sup lu(x) xEd\1
1
+
c sup If(x)
(9)
1
xE[I
for some constant C, depending only on diam
~2, AO and
sup Ibex) I.
o
xEr2
Next we formulate a few results which can be obtained from a potential theoretic approach, via the so-called Schauder estimates. In order to formulate these results we need to introduce some function spaces. 2.3.7. Definition. Let [I eRn be an open bounded domain, and consider the space of restrictions to ri of Coo-functions u defined in an open neighbourhood of ?2. Let k norms : k I j=O
32
E
IN and
°
sup sup XE[llsl=j
~ a ~ 1;
1DSu(x) 1
consider the following norms and semi-
(10)
(here
Q fJ
E
Nn ,
H (u) a
lIull k
=
lsi lu(x)-u(y) sup x,yEQ IIx-ylia xfy
I
(11 )
(12)
,a
'The completion of this space of COO -functions with respect to the norms 11.11 k' respectively I .lI k gives us the Banach spaces Ck(fi) respectively Ck,a(fi). k ,a C (fi) is the space of functions u in Q having partial derivatives up to order k, which can be continuously extended up to fi. Ck,a(fi) is the space of functions u E Ck(fi) whose k-th derivatives are uniformly Holder continuous in fi with exponent a. For a = 0, the space Ck,O(fi) coincides with ck(fi). For a = 1, Ck ,1(fi) is the space of functions u E Ck(fi) whose k-th derivatives are uniformly Lipschitz continuous in fi. By Ck,a(Q) we will denote the space of those functions in Ck(Q) whose k-th derivatives are locally Holder continuous with exponent a, i.e. whose k-th derivatives are Holder continuous in compact subsets of Q. C~,a(Q) is the space of those u E Ck,a(Q) having compact support in Q. 2.3.8. Defini tion. A bounded domain Q in JRn and its boundary ClQ are said to be of class Ck,a (kEN, O«;a«;1) if at each point x E ClQ there is a neighbourhood B of x inJRn , and a one-to-one mapping ~ from B onto D eJRn such that n Q) elRn+
(i)
~(B
(ii)
~(B n ClQ) e ClJR~
=
{xElRnl xn > O}
(iii) ~ E Ck,a(B) 2.3.9. Definition. Let Q be a bounded Ck,a-domain inJRn , and ¢ : aQ ~JR. Then ¢ is said to belong to the class Ck,a(ClQ) if, at each point x E ClQ, -1 k a n ¢o~ E C ' (D n CllR+), where ~ and D are as in definition 2.7.8. 2.3.10. Theorem. Let ~ be a Ck , a -domain in JRn (k;;' 1) and let Q' be an k a open set containing Q. Suppose ¢ E C ' (3Q). Then there exists a function 33
k a k a CO' (Sl') such that q, = ¢ on 8Sl. Also, if u E C ' (~), then there exists a function w E C~,a(~,) such that w = u in ~ and q,
E
0 where C = C(k,~,~'). So we can always consider a boundary function ¢ E Ck,a(8~) as an element of Ck ,a(f2). This is what we will do in what follows.
2.3.11. Theorem. Let Sl be a Ck,a domain inlRn , with k;;;, 1, and let S be a
bounded set in ~,a(~). Then S is precompact in cj,S(~) if j+S < k+a. That means, the imbedding
is compact. 0 Now we state the two main results of the Schauder theory of elliptic partial differential operators. We assume a > O.
2.3.12. Theorem. Let ~ be a bounded C2 ,a domain inlRn , and let L be strictly elliptic in ~, with coefficients belonging to Ca(~). Let u E C2,a(~) be a solution of Lu(x) = f(x)
XE~
u(x) = ¢(x)
X E
(13) 8~
lIull 2 ,a ';;:;C(IIUII O +II¢1I 2 ,a +lIfIIO ,a )
for some constant C depending only on n, a, ~ and the coefficients of L. Inequality (14) gives the so-called global Schauder estimate for 2 a _. C ' (Sl)-solutions of (13).
34
(14) 0
2.3.13. Theorem. (Existence). Let n be a bounded C2,a domain inRn, and let L be strictly elliptic in n, with coefficients belonging to Ca(~) and c(x) ~ 0 a 2 a _. for all x En. Then for all f E C (n) and for all q, E C ' (n) the Dirichlet problem (13) has a unique solution u E C2,a(~). D Using the preceding results, we can prove the following Fredholm alternative for elliptic operators . . 2.3.14. Theorem. Let n be a bounded C2 ,a domain inRn , and let L be strictly elliptic in n, with coefficients belonging to Ca(~). Then either: (a) the homogeneous Dirichlet problem :
Lu(x)
XEn
= 0
(15)
u(x) = 0
has only the trivial solution u = 0, in.which case the nonhomogeneous problem (13) has a unique C2,a(~) solution for each f E Ca(~) and for each q, E C2 ' a (n) ; or (b) the homogeneous problem (15) has nontrivial solutions which form a finite dimensional subspace of C2 ' a (n). Proof. Letting u = q,+v, problem (13) becomes equivalent with Lv(x)
=
XEn
f(x) - Lq,(x)
vex) = 0
X E
,m
So we can restrict attention to the Dirichlet problem with homogeneous boundary conditions Lu(x)
=
XEn
f(x)
(16)
u(x) = 0
X E
an
Introducing the Banach space B
=
2 a {uEC ' (n)
I u(x)
=0, VXEan}
35
we can consider L as a bounded linear operator from B into Ca(~), and the problem becomes that of determining the solvability of the equation Lu = f with f E Ca(~). Let 0> sup c(x), and L = L-0. By theorem 13 the equation L u f has a XE~
unique
0
0
solution u E B. Using theorem 12 and theorem 6 we find:
-1
a-
2a-
That means that the operator L0 : C (~) 7 B is bounded. Since C ' (~) (and so also B) is compactly imbedded in Ca(~), we can consider L-o 1 as a compact linear operator from Ca(~) into itself. Now let u E B be a solution of (16). Then u E Ca (~) and Lu = L u + au = f a
(17)
'
which, after application of L- 1 , gives a
( 18) Conversely, let u E Ca(~) satisfy (18). Since R(L- 1) = B, this implies a that in fact u E B. So we can operate on (18) with L, which gives (17). We conclude that the problem of finding u E B satisfying (17) is equivalent to that of finding u E Ca(~) satisfying (18). Now L- 1 is a compact operator in Ca(~). Applying the results of section o 2.1 to equation (18), we see that (18) has a solution u E Ca(~) if and only if = 0 for each u* E (Ca (~))* such that o
u* + 0 (L -1 ). u* = 0 • o
(19)
Since = «L- 1)*u* ,f>, and since u* has to satisfy (19), the condia 0 tion on f becomes: = 0 for each u* E (Ca(~)) satisfying (19). The space of such u* is finite-dimensional, with a dimension equal to dim{uECa(~) I u+aL-o 1u=O}, which, in turn, is equal to the dimension of the solution space of (15). This proves the theorem. 0
36
2.3.15. Corollary. Under the conditions of theorem 14, the set I of real A for which the homogeneous Dirichlet problem Lu(x) - AU(X)
=
0
X E
rl
X E
Clrl
(20)
u(x) = 0
has a nontrivial solution is countably infinite, and has no finite accumulation point. Furthermore, if A rF I, then the problem Lu(x) - AU(X)
=
f(x)
X E
rl
X E
Clrl
(21)
u(x) = cjJ(x)
has a unique solution u E C2 ,a(~) for each f E Ca(f,i) and ¢ E C2,a(~); moreover, this solution satisfies (22)
lIull?. . . ,0.. ';;;C(lIcjJL,.::..,0. +IIfIlO) ,cx with C independent of u, f and cjJ. Finally, by theorem 4 or theorem 13, Z c ]-oo,a[ for a = sup{c(x) I xES'lL 0 Example : for the Laplacian we have Ie] -00,0 [ . 'The foregoing shows that L : C2 ,a(fi) -+ Ca(fi) is a Fredholm operator.
A better description of R(L) than the one given in the proof of theorem 14 can be obtained using the so-called L2_ or Hilbert space-approach to the Dirichlet problem. We will briefly review this theory, mostly concentrating on the case of the Laplacian (L = 6). 2.3.16. Definitions. Let rl be a bounded open dana in inRn , and u,v E Lioc(rl). For a multi-index a, we say that Dau = v in the w~aQ ~~n6~, and that v is the a-th w~aQ d~~vativ~ of u, if v¢ E C~ (rl)
(23)
We denote by Wk(rl) the space of functions which are k-times weakly differentiable.
37
Let Wk,PC~) be the space of functions u E Wk Cr2) such that D~ E LP(~) for all lal < k. This is a Banach space, using the norm : lIull
=
Wk,p
cf
Y IDau IPdx) liP.
(24)
r2 laT~
We denote by W~,p(~) the subspace of Wk,p(~) obtained by the completion of
I
C~(r2) with respect to the norm (24).
In the case p = 2 one also uses the notation HkC~) = Wk ,2CQ) and H~C~) = W~,2Cr2). These are Hilbert spaces, with inner product:
f (25)
Finally, we will write C·,·) for C·',)O' and 1I.lI k for 11.11 k H C~) There exist a whole scale of imbedding theorems between these so-called Soboiev-~pac~ (see e.g. Adams (11). We just mention the following results; an elegant proof, using periodic extensions, can be found in Bers, John and Schechter [ 22 1. 2.3.17. Theorem. Let u
E
H~(~) and
0
< m< k -
l'
Then u
E
CUCfi), and
o
2.3.18. Theorem. Let 0
~
j
(26)
< k. 1nen H~(~) is compactly imbedded i~ HlCr2).
0
2.3.19. Theorem. The norm 1I.lI k in H~(~) is equivalent with the norm (27) Proof. It is sufficient to prove the inequality (28) for each ¢ E C~(~). Considering ¢ as the product of ¢ with a test function which is equal to 1 in ~, we have, for x E ~ :
38
I
Iri~.12,;;;C Clx. 1 1
1cp(x) 12
1
-00
,;;; c 1
rroo
2 i 1.lc£...1 Clx.1 dx.1
-00
-00
1.lc£...1 Clx.1 2dx 1.
by the Schwarz inequality. Integration over
I I I
2.3.20. Lerrnna. Assume that CljU
o
-~(x)
ClV J
Cl~
Vx E
~
gives (28).
k
is of class C . If u Cl~
o ,;;;
k
0
k-1 -
E HO(~)nc
(~),
then (29)
j ,;;; k-1
where Cl/ClV is the derivative in the direction of the outward nonnal to
Cl~.
0
2.3.21. Lerrnna. Let Cl~ be of class Ck , and u E ck(~). If also (29) is satisfied, then u E H~(~). 0 2.3.22. The generalized Dirichlet problem. Let u cal solution of the Dirichlet problem : -i'lll(x) = f(x)
X E ~
u(x)
X E
E
1C2 (~)nc (~) be a classi-
(30) =
0
Cl~
Multiplication by some ¢ E C~(~), and integration over ~ gives us B(¢,u)
V¢ E C~(~)
(¢ ,f)
(31)
where n
B(¢,u)
=
I
i=1
(D. ¢,D. u) • 1
1
(32)
However, (31) makes sense for all u E H1(~). We will consider any u E H1(~) satisfying (31) as a generalized solution of -6U = f in ~. As for the boundary conditions, if u is a classical solution as described above, then it follows from lerrnna 21 that u E H6(~). So we can give the following definition :
39
A function u E H6(~) satisfying (31) is called a gen~zed ~otution of the Dirichlet problem (30). The problem of finding such a function is called the genenalized V~ehlet p~oblem for (30). 2.3.23. Theorem. Let a~ be of class C1 . If u E C1 (n) nc2(~) is a classical solution of the Dirichlet problem (30), then u is a generalized solution of the Dirichlet problem. Conversely, if u is a generalized solution of the Dirichlet problem (30), and if u E c2(~)ncO(n), then u is a classical solution of (30). 0 It is clear that condition (31) implies that the same relation remains valid for each ~ E H6(~)' The following estimate forms the basis for the proof of the existence of generalized solutions. 2.3.24. Theorem (Garding's inequality). B(u,v) , as defined by (32), is a bounded coercive bilinear form over H6(~) (cfr. theorem 1.13), i.e., there exists some v > 0 such that Vu E
H~(~)
(33)
2 Proof. This follows immediately from theorem 19 and B(u,u) = lull'
0
2.3.25. Existence theorem. For each f E L2(~) there exists a unique , solution of the generalized Dirichlet problem for (30). That means, there exists a unique u E H6(~) such that B(v,u)
=
(v,f)
(34)
Proof. The linear functional F(v) = (v,f), v E H6(~)' is bounded. The theorem then follows fram the Lax-Milgram theorem (theorem 1.13). 0 The problem, and the corresponding existence result can also be formulated as follows. Fix same u E H6(~)' The map v # B(v,u) from H6(~) intoR is then bounded and linear. By the Riesz representation theorem we can associate with this functional a unique element of H6(~)' which we will denote by Lu. Thus L : Hb(~) ~ H6(~) is such that 40
B(v,u)
=
(v ,Lu) 1
1 ¥u,v E HO ($1) .
(35)
Since B is bounded, L is a bounded linear operator over H6($1). Let now f E H6 C$1}. By the Lax-Milgram theorem and the coercivity of B, there exists a uni~le u E H6($1} such that B(u,v) = (v,f)l for all v E H6($1); moreover, lIul1 1 ,,;; ~II fill. Since then Lu = f, this means that the operator L defined by (35) has a bounded inverse L-1 : H6 ($1) -+ H6 ($1) • With each f E L2 ($1) we can associate a uni~e element n(f) E H6($1) such that (v,f) = (v,n(f))l
(36)
1 The map n : L2 ($1) -+ HO ($1) is linear and bounded. The generalized Dirichlet problem (34) then takes the form
Lu = n(f) and its unique solution is given by u = L- 1n(f). Remark. Similar results hold for generalized solutions of the Dirichlet problem -l'Iu (x)
- AU(X)
=
f(x)
X E
$1
X E
3$1
(37) u(x) = 0
when A ,,;; O. Indeed, the corresponding bilinear form BA (v,u) = B(v,u) - ACV,U)
(38)
satisfies also the inequality (33). Let us now consider problem (37) for general A ER. We have the following resul t. 2.3.26. Theorem. There is a Fredholm alternative for the generalized solu-
tions of the Dirichlet problem (37). More precisely: either there exists 41
for each f
E
L2(~) a unique u
E
H6(~) such that (39)
B),(V,u) = (v,f) or the problem B;\ (v,u) = 0
(40) 1
has non-trivial solutions, which form a finite-dimensional subspace of HO(~)' If {u j 1 j=1,2, ... ,p} is a base for this subspace, then a necessary and sufficient condition for (39) to have a solution is that (f,u.) = 0, for all J j = 1, ... ,po Proof. Define a linear mapping J : H6(~) ->- H6(~} by J = TIoi, where i : H6(~) ->- L2(~) is the inclusion map. Since i is compact (theorem 18), the same holds for J. Problem (39) is then equivalent with : Lu-Uu
TI(f) ,
or, since L has a bounded inverse, with (41 ) -1
Now L J is compact, and we have a Fredholm alternative for (41). ,.Equation (41) has a solution u E E6(~) if and only if (u* ,L
-1
TI (f) ) 1 = 0
( 42)
for each solution u* E E6(~) of the homogeneoos adjoint equation u* - ;\J* (L -1 ) *u* = 0 .
( 43)
Let u* E EO1 (~) be a solution of (43), and let v* = (L*) -1 u* = (L -1 )* u* . Then, since Beu,v) = B(v,u) by (32), we have for each v E H6(~) : B(v,v*) = B(v* ,v) = (v* ,Lv) 1 = (L*v* ,v) 1 = (u* ,v) 1 = ;\ (v* ,Jv) 1 = ;\ (v* ,v) = ;\ (v, v*) , 42
;\(J* v* ,v) 1 (44)
i.e. v* E H6(Q) is a generalized solution of (40).
Conversely, if v* E H6(Q) is a generalized solution of (40), then the same relations show that u* = L*v* is a solution of (43). Then we also have -1
(u* , L
'IT
(f) ) 1 = (v*, 'IT (f) ) 1 = (v*, f) .
(45)
We conclude that (41) has a solution u E H6(Q) if and only if (v* ,f)
=
(46)
0
for each generalized solution v* of (40). This proves the theorem. Remark that if v* E H6 (Q) and (v*, f) = 0 for all f E L2 (Q), then necessarily v* = O. This shows that the two possibilities given by the Fredholm alternative exclude each other. 0 2.3.27. Theorem. If 3Q is of class C2 , then all generalized solutions of 2 . 2+k k (37) belong to H (Q). If 3Q 1S of class C (k> 0), and f E H (Q), then all generalized solutions of (37) belong to H2+k(Q). If 3Q is of class COO and 0 f E COO (i'i), then also u E COO (i'i) . 2.3.28. Suppose now that 3Q is of class C2,a and f E Ca(i'i). If the Dirichlet problem (37) has a classical solution, then this classical solution is also a generalized solution. The same holds for the classical solution of the homogeneous equation -lIu(x) - AU(X) u(x)
=
0
o
X E Q
(47) X E
3Q
It follows from theorem 26 that necessarily (u,f) = 0 for each classical 2 a solution of (47). Defining the operator LA : {u E C ' (Q) I u(x) = 0, xE 3Q} .... o a C ' (Q) by LAu = -lIU-AU, this proves that ( 48)
43
However, we know from the proof of theorem 14 that dim ker LA So we can conclude that in fact we have equality in (49)
codim R(L A).
Also, each generalized solution of (47) will be a classical solution. Indeed, if not, there would be some u E H~ (n) with u ¢ ker LA' and such that (f,u) = for f E R(L A). However this contradicts (49).
°
2.3.29. Under the conditions of the preceding section, let us reconsider the
eigenvalue problem (47). Assume again that 3~ is of class C2,a Let L {A ECC I ker LA f {o} }. Since each classical solution of (47) is also a generalized solution, it follows that L c o(L), where L is -the operator defined by (35). Since L is self-adjoint, we have L cR. It follows from corollary 15 that L c 10,00[, and that for each A E L, we have dim ker LA < Also, L has no accumulation points. So we can order the elements of L as follows: 00.
We want to prove that dim ker LA1 = 1, and that there exists some u 1 E ker LA1 such that u 1 (x) > 0, Vx E ~. Our tool will be the generalized Krein-Rutman theorem 1.15. 2.3.30. For each f E Ca(~), denote by Kf the unique solution of
f(x)
-6U(X)
u(x) =
°
XE~
(50) X E
3~
Considering K as a map from Ca(~) into itself, we know already (proof of theorem 14) that K is compact. We give Ca(~) an order structure by introducing the following cone of positive elements :
This cone has a nonempty interior; for example, the function vex) _ 1, 44
Vx E Q belongs to the interior of C~(Q). Also, the operator K is positive,
as a consequence of the maximum principle (theorem 5). Indeed, if f(x) ~ 0, Vx E Q, and u = Kf, then L'lu(x) ,;;;; 0, Vx E Q. It follows from the boundary . condition in (SO) and the maximum principle that u(x) ~ 0, Vx E Q. However, there is more. Let e(x) be the solution of -u(x)
=
u(x) =
1
Vx
E Q
°
Vx
E 3Q
(52)
We will prove that K is e-positive (see definition 1.14). Let v E C~(Q) \{O}. Since vex) ,;;;; IIVII O = sup{lIv(x)1I I XEQ}, it follows from the maximum principle that
Since vex) t. 0, there is some closed ball B C Q such that vex} ~ S > 0, Vx E B. Let v 1 E C~(B)\{O} be such that 0,;;;; v 1 (x) ,;;;; S, Vx E B and let u 1 = Kv 1 • It follows from the proof of the maximum principle that, under the condi tions on the boundary which we have assumed (3Q E C2 ,a), the following holds : if u E C2 (Q) nCO (Q), L'lu(x) ~ 0, and for some Xo E 3Q Vx
E Q
then the outer normal derivative of u at Xo satisfies
3u 1
°
Applying this result on the functions u 1 and e, we find : av-(x) < and ~~(x) < for all x E 3Q. By continuity there exists a number a 1 > Osuch that
°
3
3V(u1-ae) (x)
0 for all x E U\aQ, and for all a E [0,a 1 1. Moreover, since u 1 (x) > 0 for x E Q, we can take a> 0 sufficiently small such that (u 1-ae)(x) > 0 for all x E Q\U. This shows that there is some a > 0 such that ae ~ Kv. So K is e-positive. Since A E L if and only if 1/A is an eigenvalue of K, theorem 1.15 gives us the following result. 2.3.31. Theorem. Let a~ be of class C2,a, and (53) Then A1 is a simple characteristic value (54)
1 ,
and ker(K - ~l I)n = ker(K - A11 I)
Vn
:? 1 •
(55)
E Q •
(56)
Also there exists a u 1 E ker LAI such that Vx
Proof. Except for the last statement, the theorem follows from the generalized Krein-Rutman theorem. By the same theorem there exists some
u 1 E ker LAI such that u 1 (x) ~ 0, Vx E ~, u 1 (x) t O. But -6u 1(x) ~ 0, and (56) follows from the maximum principle. 0
=
A1U 1 (x)
2.4. THE VON KARMAN EQUATIONS
In this section we have brought together a number of results on the von Karman equations, which describe the equilibrium state for thin elastic plates subject to forces and stresses along its boundary, and to normal loading. Depending on the physical situation the equations take a slightly different form, and also different boundary conditions are possible. We consider the case of a flat plate with arbitrary shape, clamped along its edges or simply 46
supported; we are particularly interested in the cases of rectangular and circular plates. We also consider the buckling problem for cylindrical plates. 2.4.1. The von KArman equations. Consider a thin flat elastic plate of arbitrary shape; let CR2 be the bounded region occupied by the plate in its undeformed state. Let the plate be subjected to compressive (or stretching) forces along its edges, which are clamped, and to some normal loading. Later we will allow the forces along the edges and the loading to depend on parameters. The equilibrium state of the plate is then described by two functions· f : +R and w : +R which satisfy in Q the following system of partial differential equations
n
n
n
2 1 (',. f = -ifW,w]
(1)
(',. 2w = [w,f] + [w,FO] + p
(2)
subject to the boundary conditions f
f y = w = wx = wy
fx
0 along aQ •
(3)
In these equations (',.2 is the biharmonic operator, and [u,v]
=
uxxvyy + U yyvxx - 2uxyvxy
(4)
These equations are obtained after a number of transformations and rescalings. Physically, F = FO+f represents the so-called Airy-stress function; Fa is that part of the stress-function produced in the plate when the normal loading is zero and when the plate is artificially prevented from buckling; f is the excess stress-function produced by the buckling of the plate. The function w represents the deflections of the plate out of its equilibrium plane, and p is a measure for the normal load (and so is a given function). Also FO is supposed to be given : it represents in fact all the external forces along the edges. We will suppose that FO is bounded in together with its derivatives up to second order.
n,
47
2.4.2. Definition. A ctao~ica1 ~otution of the equations (1)-(3) is a pair of functions (f,~), both belonging to C4 (rl) nC 1 (fi), and satisfying (1)-(3) pointwise. We want to use the same approach to the von Karman problem (1)-(3) as we used in section 3 for the generalized Dirichlet problem. Let us first state the following results on LP- and Sobolev spaces. 2.4.3. Generalized Holder inequality. Let u i i = 1, ... ,m, and
E
LP i (rl), 1 ,;;;; Pi ,;;;; +00,
m
I - -
1 •
i=l Pi m
Then IT u. E L1 (rl), and i=l l o
(5)
2.4.4. Kondrachov compactness theorem. Let rl be a bounded domain inlRn . Then the following imbeddings are compact : (i)
ifkp n
2.4.5. Generalized solutions. Suppose (f,w) fonn a classical solution of (1)-(3), belonging to C4 Crl) n c2 (fi) , and assume that the boondary em is sufficiently smooth (say of class C2). Multiplying (1) by some ¢ E C~(rl) and integrating over rl gives, after some integrations by part 48
I
n (MHM)
=
I
V
(B(u,v),w) defines a synnnetric multilinear
fonn over H : (B(u,v),W)
=
= B(u,w) ,v)
(B(v,u),w)
2.4.12. Lemma. The map C
Vu,v,w E H
(19)
H -+ H is compact and cubic Va
Em.
and is the gradient of the CCO -fllilctional (J (J(w)
o
1 = "4(C(w),w)
Vw
E H •
Vw H -+
m.
E H
(20)
defined by
o
(21)
The proofs are similar to the proof of lemma 10. Remarkt that the synnne-
try relation (19) implies that (C(w),w) = IIB(w,w) 112
Vw
E
H .
(22)
We also have :
DwC(w) . u =
1 -2B (B(w, w) ,u)
+ B (B(u,w) ,w)
Vu,w E H .
(23)
51
2.4.13. Example. Consider a circular plate subjected to a uniform compressive pressure at its boundary. We can aSSlnne that the radius is equal to 1, so we have :
We will assume that the compressive pressure is proportional to a real parameter A ;;. 0; then we can take 1
2
2
FO(x,y) = -ZA(X +y )
(24)
(If A < 0, this represents a uniform tension along the boundary). The equations take the form : 2
1
f'. . f "'-jw,wj (25)
2
f'. . w = [w, f 1 - Af'...W + P together with the clamped plate boundary conditions f
Y
o=
w '" w = w along 3Q . x Y
(26)
The operator A is defined by
fQ
=
-fQ
-fQ
cjJM
(Aw)f'...2cjJ
wf'...cjJ
E Ii
VcjJ
E
C~(Q)
\lcjJ.\lcjJ ;;. 0
VcjJ
E
C~W)
Vw
(27)
It follows that
(AcjJ,cjJ) =
fQ
SO A is a positive operator, i.e. (Aw,w) ;;. 0
Vw
E
H .
(28)
Since A is compact and self-adjoint, its spectYllffi consists of real eigenvalues converging to zero. So the characteristic values of A, i.e. those A E 1R for which the equation 52
w - AAwO
(29)
has a nontrivial solution w E H, form a sequence A1 < A2 < ... < An < ... converging to +00. Each of these characteristic values has a finite lTD.lltiplicity. We now determine some of these characteristic values. 2.4.14. Characteristic values corresponding to radially symmetric eigenfunctions. Since Q is a smooth domain, solutions of (29) correspond to classical solutions of the following boundary value problem 2
in Q
6. w + A6.W = 0
w=wx =wy
0
in
em
(30)
Since both the domain Q and the equation (30) are rotationally invariant, it follows from the representation theory for the rotation group as given in section 6 that eigenfunctions of (30) either are radially symmetric, or are linear combinations of functions of the form ak(r)coske and ak(r)sinke, for SOIre k = 1 ,2, . .. (Here (r, e) are the polar coord ina tes of the point x E Q). In the first case the corresponding characteristic value is generically simple, in the second case its lTD.lltiplicity is at least two, generically it is equal to two. In order to determine the characteristic values of (30) correspondong to radially symmetric eigenfunctions, let w = w(r) be a radially symmetric solution of (30). Then it is easily seen that ~(r) = ~;(r) satisfies 2 d
~ +.l~ + (A-2-2)~ = 0
dr 2
~(O)
r dr
=0
r
~(1)
(31) =
0 .
By the positiveness of A we only have to consider the case A > o. Putting ~(r)
= ~(~r), the equation (31) and the first boundary condition ~(O) = 0 show that ~(x) = J 1 (x), the first order Bessel function. Then the second boundary condition ~(1) = 0 takes the form
o,
(32) S3
and gives us the characteristic values of A corresponding to radially syrrnne-':, tric solutions. These characteristic values are simple and form a sequence \1 < \2 < ... < \n < ... tending to +00; \1 corresponds to an eigenfunction which satisfies w(x) > 0, Vx E Q. All other characteristic values of (30) are at least double; one can prove that they are all strictly greater than
2.4.1S. The simply supported plate. Consider again the von Karman equations for the buckling of a plate, bllt let us assume now that the plate is simply supported instead of being clamped. The equations (1) and (2) remain, but there are different boundary conditions; the general theory for this case has been described in Berger & Fife [18]. Here we will only consider the partiClllar case of a rectangtllar plate, for which the theory can be considerably simplified. Let the domain occupied by the plate in its undeformed state be {(x,y)
Q
I -t<x< t,
-1 Am1 if n> 1, and that for each m. Now Am1
=
02 i2 2 -2(m +-) 4i m
2
The fWlCtion x
+>-
x +~ attains its minirum in x> 0 at the point x x
L So
we can expect the lowest characteristic value AO to be equal to Am,l for m = [ i 1 or m = [i+ 1 ]. If i = m, then AO = A l' and AO is simple. If m, 2 . m < .~ < m+l, then Am,l = Am+1, -I if and only if i = mCm+1). In that case AO has multiplicity two. In all other cases AO is simple. The special importance of the first characteristic value follows from the next resul ts . 2.4.25. Lemma. Let H = W~,2(~), respectively as in 2.4.17. Let C : H 7 H be as in (18), respectively (46). Let w E H be a solution of (29), respectively (48), for some A. E R. Suppose also that (C(w),w) = O. 'Then w = o. Pro a f. We have in both cases that (C(w),w) = II B(w,w) II 2 (efr. (22)), so (C(w),w) = 0 implies that B(w,w) = 0, or, by the definition of B and since w is sufficiently smooth :
f [W,W]¢=o ~
Consequently : w w
xx yy
w(x)
=
-w2
xy
0
=
0
in
~
Vx
E
d~
•
59
Considering the surface w = w(x,y), this implies that the Gaussian curvature of this surface is identically zero. This means that the surface is developable and contains a family of straight lines. Because of the boundary condition on w this is only possible for w = o. 0 2.4.26. Theorem. Let 1..0 be the lowest characteristic value for the problem (29), respectively (48). Then the equation w - I..PJN + C(w)
o
(58)
has, for A < 1.. 0 ' only the solution w
=
O.
Proof. 1..0 has the following variational characterization (see Courant & Hilbert [48], Weinberger [ 234]) :
-1 = sup (Aw,w) 1..0 \\EH (w,w)
(59)
wrO
Let w t 0 be a solution of (58), for some 1..< 1.. 0 . Then (22) implies that (w- Aw,w) < O. By (59) this is only possible if A = 1..0 and w E ker LAO. But then (58) gives (C(w),w) = 0, which, by lemma 25 gives w = 0, a contradiction. 0 2.4.27. The cylindrical plate. As a last example we consider a cy~indrical plate which is simply supported, axially compressed, and subject to a radial loading. The von Karman-Donnel equations which govern the equilibrium state of such system reduce after some rescaling to : 2
1
t, f = - -2I w, w 1 + awxx ' t, 2w = [w,f] - AWxx - afxx
(60) +
p .
These equations hold in the domain the boundary condition w
=
t,w
=
0
f
t,w
and the periodicity condition 60
n = {(x,y) I -TIl<x< TIl},
0 along 8Q
together with
w(x,y) = w(x,y+2n) , f(x,y) = f(x,y+2n)
, V(x,y)
E
rt .
(62)
The constant a is proportional to the curvature of the cylinder. In our setting x measures distances parallel to the axis, while y measures distances along the circumference of the cylinder; this explains the periodicity condition (62). One can use the same approach to these equations as we did before for flat plates. The basic Hilbert space will be the closure in the W2 ,2(rt O)norm of the Coo(~)-functions which are zero along art and 2n-periodic in y. (Here rto = {(x,y) I-nt<x 0, ~mn is a characteristic value of ponding to the eigenfunctions :
A
of multiplicity two, corres-
(72)
and m,n sin 2rr;,7v (X+1TJI,) sin ny .
C
(73) .
The constants Care nOllllalization constants. m,n 2.4.29. Bibliographical notes. There exists an extensive literature on the buckling problem for plates. An account of the physical motivation for the von Karman equations can be found in Landau and Lifschitz [139J and Leipholz [ 142 ]; the basic ideas leading to these equations were introduced by von Karman [ 107 ]. A general theory of the von Karman equations can be found in Berger and Fife [ 17] ,[ 18]. Further references are Ambrosetti [6], Antman [ 8 I, Bauer, Keller and Reiss [ 12],[ 14], Berger [20],[ 21], Dickey [ 58], Fife [62], 62
Friedrichs and Stoker [68], Keener [109], Keener and Keller [110], Keller, Keller and Reiss [113], Knightly [121],[122], Knightly and Sather [123],[124], [125],[ 126],[ 127], Koiter [131], Magnus and Poston [152], Matkowsky and Putnick [158], Reiss [181], Sather [190 ],[ 191], Shearer [206], Wolkowisky [236]. Recently a very nice discussion of the von Kannan equations has been given in the lecture notes of Ciarlet and Rabier [246]. A somewhat different approach can be found in Duvaut and Lions [60]. 2.5. GROUP REPRESENTATIONS AND EQUIVARI~"IT PROJECTIONS This section contains some general results on representations of groups over Banach spaces. Most of these results are taken from Rudin [184] (in particular sections 5.11 to 5.18), where full proofs can be found. Finite-dimensional representations will be considered in the next section. 2.5.1. Definition. A topofogi~al g~oup is a group G on which a topology is defined such that : (i) every point of G fonns a closed set; (ii) the map cjl
GxG+G, (s,t) +>-cjl(s,t)
st -1
(1)
is continuous. The topology of G is a Hausdorff topology, and is completely determined by any neighbourhood base of the identity element e E G. 2.5.2. Definition. Let X be a topological vector space, and G a topological group. A ~ep~~entation 06 G ove~ X is a map r : G + L(X) such that : (i)
r is a group homomorphism : r(e)
I
rest)
res) ret)
Vs, t
E
G
(2)
(ii) r is strongly continuous lim r(t)x = x
Vx EX.
(3)
t+e
63
2.5.3. Lemma. Suppose that X is a Frechet-space, G a locally compact topological group, and r : G +. LeX) a representation of Gover X. Then the map (s,x) ++ r(s)x from GxX into X is continuous. Proof. For each x E X the map s ++ r(s)x from G into X is continuous, by the definition of a representation. Fixing a compact neighbourhood U of So E G, this implies that {r(s)x I s E U} is bounded in X, for each x E X. It follows from the Banach-Steinhaus theorem that {res) I s E U} forms an equicontinuous set of bounded linear operators. This proves the lemma. 0 2.5.4. The groups used in this book will in general be ~ompact topological groups. On such groups it is possible to define an invariant measure; in order to introduce this me.asure, we need some concepts from integration theory. If X is any set, then a o-atgebna over X is a collection L of subsets of X, such that (i)
X·E L
(ii)
A E L ~ AC E L ;
(iii) An
E L ,
Vn
E N
~
u
nEN
A
n
E L
A set X together with a o-algebra L over X is called a me~unable ~pa~e (X,L). In particular, if X is a compact or locally compact Hausdorff space, then the smallest o-algebra LB containing all open subsets of X, is called /
the o-algebra of BOfl.u ~ w A po~-i;ttve me~uJte over
in X. (X,L)
is a map].l
L + [0,00]
which is cruntably
additive : A.
J
E L ,
A. nA. 1
J
= ¢ ,
i 'f j
~
].l( u
j EN
A.) J
Such positive measure is &~nite if ].l(X) < 00. In case ].l(X) = 1, ].l is called a p~obab~y m~uJte. In case X is a locally compact Hausdorff space, a po~-i;ttve Bo~u m~uJte is a positive measure over (X,LB)' Such Borel measure is
~e.gulM
if
sup{].l(K) I KcE compact} in£{ ].l(G) lEe G open} 64
2.5.5. Definition. Let \1 be a positive measure over a measure space (Q,L), X a locally convex topological vector space, and f : Q -+ X a mapping such that x*of E L'(\1), for each x* E X*. If there exists a vector x E X such that x* (x)
f
=
Vx* E X* .,
(x* of)d 11
(4)
Q
then x is called the PeA.:U.J.,-iVl:t!2gJt.af on f ovM Q wi:t:h ItVlP!2C.:t :to :t:h!2 m!2aJ.>Wl.!2 11 ; we use the notation : x = f
(5)
fd\1. Q
2.5.6. Theorem. Suppose X is a Frechet space, and \1 is a Borel probability measure on a compact Hausdorff space Q. Tnen the integral JQ fd\1 exists for every continuous f : Q -+ X. Moreover,
JQ fdll E co
f(Q) ,
the closed convex hull of f(Q) in X.
0
2.5.7. Let G be a compact topological group. Then we denote by C(G) the Banach space of all complex-valued continuous functions on G, with the supremum norm. For given s E G, we then define L : C(G) -+ C(G) and s R C(G) -+ C(G) by : s
(1, f) (t)
s
= fest)
(R f) (t) = f(ts) s
VtEG , Vf E C(G) .
(6)
2.5.8. Theorem. On every compact topological group G there exists a unique Borel probability measure m which is left-invariant, in the sense that :
J
JG fdm = G(Ls f)dm
Vs E G , Vf E C(G) .
(7)
This measure is also right-invariant
f G fdm =
JG(Rs f)dm
Vs E G , Vf E C(G) ,
(8)
and it satisfies the relation 65
fG f(t)dm(t)
=
f
f(t- 1)dm(t) G .
Vf E C(G) •
This measure is called the HaaJl me.a6UJLe. on G.
(9)
0
2.5.9. Theorem. Let X be a Frechet space, and Y a closed subspace of X, having a topological complement. Let G be a compact topological group, and r : G ~ L(X) a representation of G over X, such that r(s)(Y) c Y
Vs E G •
Then there exists a continuous projection Q of X onto Y which is e.quA-vaJuaVLt with respect to r r(s)Q = Qr(s)
Vs E G •
(10)
When P E L(X) is a continuous projection with R(P) = Y, then one can construct an equivariant projection Q from P by
f
Qx = G res
-1
)pr(s)xdm(s) ,
Vx EX,
(11 )
where m is the Haar measure of G, and the integral is defined in the sense of definition 5. 0 2.5.10. Example. Let G be a finite group G
{s. l
I i = 1 , ..• ,N}
( 12)
•
Using the discrete topology, G becomes a compact topological group. The corresponding Haar measure is given by m(E) = ~ x (number of elements in E)
VE E G •
( 13)
Under the hypotheses of theorem 9, the formula (11) for an equivariant projection takes the form : 1
Q =N 66
N
I
i=1
1 r(s~ )pr(s.) l
l
(14)
For this particular case the proof is much easier than for the general situation of theorem 9 (see Vanderbauwhede [225]). ·2.5.11. Example. Let X be a Hilbert space, G and r G + L(X) a unitany representation : Vs
E
a compact topological group,
G.
(15)
If Y is a closed subspace of X which is invariant under the representation (i.e. such that r(s)(Y) c Y, Vs E G), then the orthogonal projection Q of X onto Y is equivariant with respect to r. Proof. Let y
E
Y,
Z E
Y1 and s
E
G. Then
= = = 0 , -1
1
since res )y E Y. This shows that also Y is invariant under r. For each x E X we have: r(s)x = r(s)Qx
+
r(s) (I-Q)x .
Since, by the foregoing, we have r(s)Qx that Qr(s)x = r(s)Qx. 0
E
Y and r(s)(I-Q)x
E
Y
it follows
2.5.12. Example. In the subsequent chapters we will frequently meet the following situation, which is somewhat similar to the preceding example. Let X be a (real) Banach space, and : XxX -+:ffi. a continuous, synunetric and positive definite bilinear form over X. Let G be a compact topological group, and r a representation of G over X, such that = <x,y>
Vs
E
G , VX,y
E X
(16)
Let Y be a finite~imensional subspace of X, which is invariant under r. Let {e i I i = 1, .•. ,n} be a basis for Y, which we can assume to be orthonormal with respect to the bilinear form . Define a projection Q of X onto Y by
67
n
Qx =
I
i=1
<x,ei>e i
Vx EX.
(17)
Defining y1= {xEXi <x,y>=O, ¥xEY} one easily sees that Qx = 0 is equivalent to x E y1, i.e. ker Q = y1. Since, for each s E G, res) is an automorphism of X, the restriction of res) to Y is an isomorphism of Y onto r(s)(Y) C Y; we conclude that r(s)Y = Y, since Y is finite-dimensional. Then the same argument as in example 11 shows that r(s)(y1) C y1, and that the projection Q given by (17) is equivariant. We conclude this section with the following theorem, whose proof is very similar to the proof of theorem 9, as given in Rudin [ 184]. 2.5.13. Theorem. Let X be a Frechet space, G a compact topological group, and r : G + L(X) a representation of Gover X. Let Xo = {xEX i r(s)x=x, VSEG} .
( 18)
Then Xo has a topological complement in X, and Po E LeX), defined by (19)
is a continuous projection of X onto XO. Moreover Vs
E
G •
(20)
Proof. It is clear that Xo is closed. By theorem 2.1.2 it is sufficient to prove that PO' as defined by (29), is a continuous projection, and that R(P O) = XO· First, the integral in (19) exists since the map s r(s)x from G into X is continuous, for each x E X. It is clear that Po is linear. Since G is compact, the set {r(s)x i sE G} is canpact, for each x E X. By the BanachSteinhaus theorem (see Rudin [184], theorem 2.6) it follows that {r(s)isEG} is an equicontinuous family of linear operators in X. Let V be any neighbourhood of 0 in X. Choose a convex neighbourhood W of o such that WC V. There exists a neighbourhood U of 0 such that ;+
r(s)(U) C W 68
Vs
E
G.
it follows from theorem 6, the definition of Po and the convexity of W,
Po(U) ewe V This shows that Po is continuous. It is clear that POx = x for x E XO' Now remark that it follows from the . definition of the Pettis-integral that the integral commutes with bounded linear operators. Using the left invariance of the Haar measure, we have Vt
E
G , Vx EX.
This shows that R(P O) C XO; since PO(XO) = XO' we have in fact R(P O) = XO' Finally, it follows from the right invariance of the Haar measure that por(t)x =
fG
r(s)r(t)xdm(s)
This proves (20).
Pef
r(t)Pox
,Vt E G , Vx EX.
D
2.6. IRREDUCIBLE GROUP REPRESENTATIONS In this section we give an introduction to the theory of irreducible group representations, and then determine the irreducible representations of the groups SO(2), 0(2), SO(3) and 0(3). These groups will be frequently encountered in the subsequent chapters. Let us remark that we will concentrate on real representations, in contrast to the complex representations considered almost exclusively in classical handbooks on group theory (see e. g. Hamermesh [ 92], Miller [ 166], Husain [99]). Our approach to the representations of the group SO(3) is an adaption to real representations of the approach in Miller [ 166]. A nice account of the theory of spherical harmonics can be found in MUller [ 168]. We consider representations of the form
r
n G -+ LOR) ,
where G is a topological group. The natural number n is called the of the representation.
dimen6~on
69
2.6.1. Lennna. Let f be an n-dimensional representation of G, and let T E L(lRn) be non-singular. Then n
f1 : G -+ L(IR) , s
+>-
-1
f 1 (S) = T f(s)T
also defines a representation of G.
(1)
0
2.6.2. Definition. Two n-dimensional representations f and f1 of a topological group G are J.JJJYlilaJL or equivalent if there exists some non-singular T E L(lRn) such that Vs
E
G.
(2)
Similarity defines an equivalence relation among the representations of a group G. This reduces the study of all representations of G to the study of equivalence classes of representations. For compact topological groups, each such equivalence class contains an o~hogonal ~ep~eJ.Jentation, that is, a representation such that f(S) is orthogonal for each s E G. 2.6.3. Theorem. Each representation of a compact group is equivalent with an orthogonal representation. Proof. Denote by the inner product inlRn , and by m the Haar measure of the group G. Define a positive definite, synnnetric bilinear form B : lRn xlRn -+ lR by : B(x,y) = fGdm(S)
Vx,y
E
lRn .
(3)
It follows from the invariance of the Haar measure that B(f(t)x,f(t)y) = B(x,y)
Vt
E
G , Vx,y
E
lRn
(4)
We have B(x,y) = , for some synnnetric posltlve definite A E L(lRn). Also, there is a synnnetric positive definite T E L(lRn) such that T2 = A. Then B(x,y) = , and (4) can be written in the form : -1
-1
= <x,y> 70
Vt
E
G , Vx,y
E
lRn
(5)
thiS shows that Tr(t)T-1 is orthogonal, for each t E G.
0
2.6.4. Definition. A set M = {Mk I kE K} C LORn) of linear operators over:nf is said to be fLeduc.ible if there exists a subspace V of]Rn such that (i) 0 < dim V < n ; (ii) Mk(V) C V , Vk E K If M is not reducible, we say that M is If JIf =
N
I
ffi
j=l
(i)
(ii)
0
..)
=
~
=V
Vs
E
G.
(7)
V given by the same lemma satisfies
r(s)v*(u,>..)
Vs E G , V(u,>..) E UxwO •
(8)
Proof. Let U, V, Wo and v*(u,>..) be as in lemma 1.3. First, we can shrink U and Wo in such a way that r(s)v*(u,>") E V for each (u,>..) E UxwO and for each s E G. This follows from the fact that r(s)v*(O,O) = 0 for all s E G, the compactness of G, the continuity of V* and lemma 2.5.3. The same lemma also implies that we can find a neighbourhood U' c U of 0 in ker L such that r(s)(U') C U for all s E G. Then let 97
U,
=
u sEG
r(s)(u')
C
u
v,
=
u sEG
r(s) (V)
.
= U1 and r(s)(V1 ) = VI for each s E G. Moreover, u rcs)(UxV)cQ. Let us show that for (u 1 ,v 1 ,A) E U1xV1xwO the SEG tion (1.4a) is satisfied if and only if v 1 = v*(u l ,I\). Let (u l ,v 1 ,1\) E U1xV1xwO be a solution of (1.4a). Then v 1 = r(s)v for
It is clear that r(s)(U 1)
U1xV1
C
some s
E
G and some v
E
V. It follows from the equivariance of M and Q that
also (r-1(s)u 1 ,v,I\) E UxVxwO is a solutionof (1.4a). 111en lemma 1.3 implies -1 -1-1 that v = v*(r (s)u 1 ,1\) and v 1 = r(s)v*(t (s)u 1 ,:\). Since r (s)u 1 E U and 1\ E wo' it follows that vI E V. Then the result follows from lemma 1.3.
Moreover, (8) follows from the foregoing and the fact that (r(s)u 1 ,r(s)v*(u 1 ,1\),),)
E
01xV1xwO is a solution of (1.4a) for each s
E
G,
and each (u l ' 1\) E 01 xwO' lJ Prom now on we will assume implicitly that the neighbourhoods U and V satisfy (7). 3.3.6. Theorem. Assume (H). Then the mappings x * : UX()I O -7- X and F : UxwO -7R(Q), defined by (1.11) and (1.12), are equivariant, i.e., we have for all s E G and all (u,/\) E UxwO : x *(r(s)u,/\)
r(s)x *(u,/\)
(9)
and F(r(s)u,/\) = ~(s)F(u,l\) Pro (5).
0
(10)
£. (9) follows from (8), while (10) follows from (8), (2.7) and 0
3.3.7. Conclusion. As a result of the foregoing we may conclude that the bifurcation equation HU,I\)
=
0
obtained from (1) by the Liapunov-Schmidt reduction inherites the symmetry properties of the original equation (1), provided the projection operators used for the reduction are equivariant with respect to the symmetry
98
3.3.8. Bibliographical note. The Liapunov-Schmidt reduction for symmetric equations as given in this section is a generalization of the treatment in Vanderbauwhede [ 225], where attention was restricted to finite groups. Some. what similar results were obtained by Loginov and Trenogin[ 148], who introduced the commutation relations (4) and (5) as a supplementary hypothesis, and by Sattinger [195], who used particular projections P and Q, defined in tenns of dual bases. The possibility to choose P and Q such that (4) and (5) ate satisfied was recognized by Sattinger in [ 197], where also the formula (2.5.11) is given. 3.4. SYMMETRIC SOLUTIONS 3.4.1. In this section we show that there is a one-to-one relation between the solutions of M(x,>.)
=0
(1)
which are invariant under the symmetry-operators {r (s) try-invariant solutions of the bifurcation equation
I s E G},
and the symme-
(2)
F(u,>.) = 0 •
We assume the hypotheses (H) of the preceding section, and use also the notation of that section. Let Xo and Zo be the subspaces of X, respectively Z, given by (2. 11) and (2.12). By theorem 2.5.13 we can define projections Po and QO on Xo and ZO' respectively, by : Vx E X
(3)
Vz E Z •
(4)
and
Q z = J f(s)zdm(s)
o
G
3.4.2. Lemma. We have (5)
99
and (6)
~ = GoQ •
Proof. Let x E X. Then
PPox = pfGr(S)Xdm(S)
fGPr(S)Xdm(S)
fGr(S)Pxdm(S)
= POPx . An analogous proof holds for (6).
3.4.3. Lennna. Let u
E
0
unXO and A E wOo Then (7)
and (8)
F(U,A) = GoF(U,A) •
Proof. We have, for u X*
E
(U,A) =x* (r(s)U,A)
un XO' A E
Wo
and
S E
G
r(s)X* (U,A)
and F(U,A)
= F(r(s)U,A)
This implies (7) and (8).
r(s)F(u,A)
0
3.4.4. Theorem. Assume (H). Let U, V, wO' X* and F be as in lemma 3.5 and theorem 3.6. Then the following statements are equivalent for each x E UxV, U E
U and A E x
E
Xo
(ii) u
E
Xo
(i)
Wo :
Px = u and M(X,A) = 0 '
X =
x *(U,A)
and (9)
100
pr 0 0 f. Let (i) be satisfied. TIlen POu = POPx = PPOx = Px = u; so UElhXO' Further, we have from theorem "1.6 that x x*(u,\) and F(u,\) = O. This implies (ii). Conversely, let (ii) be satisfied. Then, using the preceding lemma x
x *(u,\)
and
F(u,\) By
QOF(u,\) = 0
theorem 1.6 this implies (i).
0
3.4.5. Corollary. Under the hypotheses of theorem 4, let x E UxV, \ E wO' U = Px and M(x,\) = O. Then x E Xo if and only if u E XO' 0 3.4.6. Conclusion. It follows from theorem 4 and corollary 5 that, as long as we restrict to a sufficiently small neighbourhood of the origin, there is a one-to-one relationship between the solutions x E Xo of (1), and the solutions u E ker L n Xo of (2). When one wants to construct the bifurcation equations for such solutions two approaches are possible : (i)
For x E Xo we can replace (1) by (10) (see remark 2.5(ii)). The left-hand side of (10) is a map from (Si n XO) Xw into ZO' Then we can apply the Liapunov-Schmidt method to this reduced equation, using the projections P1 = PIXo and Q1zo' By lemma 2 these projections have the desired properties.
(ii) We can also construct the bifurcation equation (2) for the equation (1); when restricting attention to u E Un Xo in (2), we can replace (2) by : QOF(U,A)
=
0 ,
(11)
where the left-hand side of (11) is considered as a map from (U n XO) into R(Q) n ZO' 101
By the theory above both approaches give exactly the same bifurcation equations for the reduced problem. 3.3.7. Bibliographical note. Stokes [210] has given a somewhat different approach to the problem of the reduction of the bifurcation equations for solutions in certain invariant subspaces. A discussion of ~le relation between both methods is given in [226]. The same paper also describes a set of hypotheses which are intermediary between (H) and the basic hypotheses of [210]; starting from these intermediary hypotheses one can recover most of the results of this chapter. 3.5. APPLICATION : REVERSIBLE SYSTEMS In this section we show how some results of Hale [77] for reversible systems (also called systems with "property E") can be derived from the results of the preceding sections. 3.5.1. The problem. Consider the ordinary differential equation
x = A(t)x
+
f(t,x,A)
(1)
where ~
(i)
x
(ii)
A: R ~ L~) is continuous and 2n-periodic;
E
(iii) f : R xRn x w ~ ~ is continuous, and C1 in (x, A); here w is a neighbourhood of 0 in some Banach space A ; (iv)
f(t,x,A) is 2n-periodic in t;
(v)
f(t,x,O) = 0
V(t,x) EJRxRn.
We will be looking for 2n-periodic solutions of (1). 3.5.2. The functional formulation. Let us introduce the Banach spaces (of 2n-periodic functions) X and Z, and the linear operator L : X~ Z as in section 2.2. Defining the nonlinear operator N : Xxw ~ Z by 102
N(X,A)(.) = f(.,xC.),A)
(2)
we can reformulate the problem as follows Find (X,A) E Xxw such that Lx = NCX,A)
(3)
It is easy to see that N is once continuously Frechet-differentiable, and
N(x,O)
Dx N(x,O) = 0
°
Vx EX.
(4)
We use the results of section 2.2. Let P and Q be the projections defined by (2.2.19) and (2.2.20). Also, let K : R(L) ~ ker P be the pseudo-inverse of L; i.e., for each z E R(L), Kz is the unique 2n-periodic solution of
:ic = A(t)x
+
z(t)
(5)
also satisfying Px = O. The operator K(I-Q) is then a bounded linear operator from Z into X. Let us define the following isomorphisms betweenJRP (p = dim ker L) and ker L, respectively RCQ) : JRP ~ ker L
a ...,. a
l a.¢. l a.l/!. i=l i=l
IjI
: mP
~ R(Q)
a ...,. ljIa
l
l
l
l
(6)
.
Using an adapted version of the Liapunov-Schmidt method, we can prove the following result. 3.5.3. Ineorem. For each R> 0, there exists a AO = AO(R) > 0 such that, for each (a,A) E JRPXA with I all .;;;; Rand IlAIl .;;;; AO(R) , the equation x = a
+
K(I-Q)N(x,A)
(7)
has a unique solution satisfying I (I-P)xll .;;;; R. This solution x *(a,A) is a C1-function of (a,A) and x*(a,O) = a. 103
~reover, if for I all " R and 11 All " AO (R) we define G(a, A) E lRP by
G(a,A) = ,¥-l[QN(x*(a,A),A) 1
(8)
then, for each (a,x,A) E lRPxXxA satisfying I all "R, I (I-P)xll " R and IIAII " AO(R) , the following are equivalent (i)
Px
= ~a
,
and
(ii) x = x*(a,A) G(a,A)
=
Lx
= N(X,A)
and
0 •
Proof. As in the Liapunov-Schmidt reduction one proves that equation (3) is equivalent to v = K(I-Q)N(u+v,A)
(10
o = QN(U+V,A)
(10
•
Fix some R> 0, and let A(R) A E w consider the map : F(a,A) : A(R)
+
{vE X1 _P IIIVII';;;R}. Also,
X1 _P , v
#
F(a,A)v = K(I-Q)N(~a+v,A)
Choose AOCR) > 0 sufficiently small, such that : sup{lIf(t,x, A) I I tElR, I xII "(l+II¢II)R, !lA1l "AO(R)} " RIIK(I-Q) 11- 1 and sup{lI af ax (t,x, )11 I tE:ffi., IIxll 1
" ZIlK(I-Q) I
"(l+II~II)R, nAil
-1
"AO(R)}
•
This is possible because of (4). Then it is easy to see that for each (a,A) E :ffi.Pxw satisfying lIall " R and !lA1l " AO(R) we have 104
F(a,A)
A(R)
-)0
A(R)
while F(a,A) is a contraction, with a contraction constant (= 1/2) independent of (a,A) in the domain considered (i.e. we have a uni60JUn contraction). It follows that the map F(a,A) has a unique fixed point v*(a,A) E A(R) , for each (a,A) such that lIall ,,;;; R, HII ,,;;; AO(R). It is then clear that x*(a,A) = ~a+v*(a,A) will satisfy the requirements of the first part of the theorem. The second part follows then as in the Liapunov-Schmidt approach, reminding that '¥ is an isomorphism between m.P and R(Q). 0 For each A, the bifurcation equations (8) form a set of p scalar equations in the p unknowns a.1 (i = 1, ..• ,p). Also G(a,O)
o
Va
E
m.P
(11 )
while G(a,A) is a C1-function of its arguments (where defined). Now we introduce some symmetry in the problem. 3.5.4. Definition. Let g : m.xm.nxw -)om.n , and consider the first order
differential equation x
=
g(t,x,A) .
(12)
Then we say that (1) is ~ev~ib{e if there exists a symmetric S E L~) such that S2 = I and Sg(-t,SX,A) = -get,x,A)
V(t,X,A) Em. xnf x w .
One also says that system (12) has the "pMpeJL:ty E wU;h (see Hale [77]).
( 13) ~e.ope& :to
S"
3.5.5. Reversible periodic systems. Assume that equation (1) is reversible
SAC -t)
-A(t)S
Vt Em.
(14)
and
105
Sf(-t,Sx,A) = -f(t,x,A) , Vt
E
n· lR , Vx
n lR , VA
E
w •
E
( 15)
In order to introduce the fonnalism of the preceding sections, consider a two-element group G = {e,i}, where e is the identity, and i 2 = e. We define representations rand r of G by x
r(i)x(t)
Sx( -t)
Vx
E
X, Vt E lR
(16 )
f(e)z = z
rei) z(t)
-Sz(-t)
Vz
E
Z, Vt
(17)
r(e)x
=
E
lR
It is easily verified from (14) and (15) that
r(s)L
Vs
Lf(s)
E
G
and N(r(s)x,A)
Vx
r(s)N(x,A)
E
X, VA
E
w, Vs
E
G.
That means: (3) is equivariant with respect to (G,f,r). Remark that rand do not coincide on X. According to the example 2.5.12 the projections P and Q satisfy the commutation relations (3.4) and (3.5). So ','le are in a position to apply the results of the preceding sections. 3.5.6. Since r(i)(kerL) = ker Land f(i)(R(Q)) singular (pxp)-matrices Band 13 such that :
r
R(Q) there exist non-
p S<jJ.(-t) = J
I
Bk·~(t) J
j
1, ... ,p , Vt
E
lR
( 18)
I l\'1f\(t) J
j
1, ... ,p , Vt
E
lR •
( 19)
k=l
and p -Slj,I. (-t) J
k=l
This implies f(i)cPa
106
..) E WxwO of equation (2) satisfy r(s)x = x
o
(5)
3.6.3. Theorem. Assume (H) and (6)
Zo C R(L) ,
i.e. (7)
Let U be the neighbourhood of the origin in ker L given by theorem 4.4. Then (x*(u,>..),>..) is a solution of (2), for each Cu,>..) E (unXO) xwO; also, each of these solutions belongs to XO' 112 , I
proof. It follows from (7) that, for z E R(Q), QOz tion equation (4.11) is trivially satisfied. D
O. So the bifurca-
3.6.4. Example. The foregoing theorems are abstractions of the results in example 5.11. Indeed, in this example the restriction of res) to ker L is the identity, for each s E G; also, the restriction of rCs) to R(Q) equals minus the identity. For this example the conditions (1) and (6) are satisfied. 3.6.5. In the next theorem we will make a more detailed hypothesis on the symmetry properties of the elements of ker L. In order to formulate the hypothesis, consider a closed subgroup G' of G, together with the corresponding invariant subspace
Xb
= {xEX I r(s)x=x, YsEG'}
(8)
and projection operator (9)
(m' is the Haar measure of G'). We will assume that ker L n Xb generates ker L by the action of the symmetry operators; more precisely : (R) For each u E ker L one can find s E G such that
r(s)u E ker LnXb .
(10)
~otheses
similar to (R) will play an important role at several places in the following chapters, and more in particular in the main result of chapter 4.
3.6.6. Theorem. Assume (H) and (R). Let U, V and Wo be the neighbourhoods given by lemma 3.5. Let (X,A) E (UxV)xwO be a solution of (2). Then there exists some s E G such that r(s)x E
Xb .
(11)
113
Proof. Let u = Px; then x = X*(U,A) and F(u,A) = O. By (R) we can find s E G such that f(s)u E ker L n XO. Then f(s)x
=
f(s)x *(U,A)
=
x *(f(S)U,A) ,
and, for each t E G' : f(t)f(s)x = x*(f(t)f(s)U,A) = X*CfCS)U,A) This proves the theorem.
f(s)x .
0
3.6.7. Corollary. Under the hypotheses of theorem 6, all solutions
(X,A) E (UxV)xwO of (2) have the form x = rcs)xO '
with s E G, and with (XO,A) a solution of (2) such that Xo E XO. Conversely, of (XO,A) is such a solution, then (f(S)XO,A) is also a solution of (2), for each s E G. 0 This corollary allows us to restrict attention to solutions of (2) belonging to the subspace XO. By theorem 4.4 this gives a reduction of the bifurcation equation. 3.6.8. Example : subharmonic solutions. Consider again the 2n-peri~dic equation
x = A(t)x
+
f(t,x,A) ;
( 12)
making the same assumptions as in subsection 3.5.1, we will this time however look for ~ubhanmonLe solutions of (12). These are solutions having a least period which is an entire l1Ultiple of the basic period 2n of (12). Using the adapted Liapunov-Schmidt reduction for (12), (as in section 5) am theorem 1, we obtain the following negative result: Let m EN\{O} and suppose that each 2nm-periodic solution of
:ic 114
=
A(t)x
(1
is in fact 21T-periodic. Then each 2mn-periodic solution of (12) with sufficiently small, will also be 21T-periodic. For the proof, we replace 21T by 2nrn in the treatment of section 5. For example, we consider the spaces x(m) and Z (m) of continuously differentiable, respectively continuous, 2nrn-periodic functions. Consider also the group G = l (mod m) and its representation (f(p)z)(t) = z(t+21Tp)
Yt E JR , Yz E Z(m)
(14)
p = 0,1, ... ,m-1 . Defining L(m) : x(m) + z(m) formally as before, the hypothesis shows that r(p)u = u for each u E ker L(m) and each p = 0,1, ... ,m-1. The result follows from theorem 1. The region where the result is valid is the same as the one given by theorem 5.3 for the validity of the Liapunov-Schrnidt reduction. 3.6.9. Example: conservative oscillation equation. Consider the problem of finding 21T-periodic solutions of the autonomous scalar equation :
x + x = g(x,A)
(15)
where g : JRxJ\ +JR is a C1-function, satisfying g(O,O) = ~~(O,O) = o. Of course there is no reason whe we should only consider 21T-periodic solutions of (15). However, if we want to find periodic solutions of (15) having a period near to 2, then by a time rescale (as in example 5.12) the problem can be reduced to that of finding 21T-periodic solutions of another equation, which is of the form (15) if we include the (unknown) rescaling parameter in the new parameter A.
Let
z = {z :JR+JRI z X = {x E Z I x is L : X
+
is continuous and 21T-periodic} of class C2}
x-»Lx=x+x
Z
and
N
XxJ\
+
Z
(x,A) -» N(x,A)(t)
g(xCt) ,A) .
115
Our problem can be written in the form
= N(X,A)
Lx
We have ker L
span{cos(.),sin(.)}
and R(L) = {z E Z
2n
I f0
J2n
z(t)cost dt = 0 z(t)sint dt = OJ .
( 18)
Let G = 0(2), and define a representation of Gover Z as follows (f(a) z)(t)
z(t+a)
Va EJR, Vz E Z, Vt EJR,
( 19)
Vz E Z, Vt E JR •
(20)
and (f z)(t) = z(-t) a
(See subsection 2.6.13 for the notations f(a) and fa)' The restrictions of f(a) and fa to X define a representation of 0(2) over X, which we also denote as f. The mappings Land N are equivariant with respect to (G,f); the hypothesis (R) is satisfied for G' = {<jJ(O)=I,a}, and Zo is the space of continuous, 2n-periodic and even functions. An application of the6rem 6 and corollary 7 gives the following conclusion Each solution (X,A) E XxA of (15), with IIxll and I All sufficiently small, becomes an even function of t after an appropriate phase shift; i.e. xC -t+a)
x(t+a)
Vt
E
JR ,
for some a EJR. When solving (15) for small 2n-periodic solutions it is sufficient to consider even 2n-periodic solutions. The blfurcation equations for (16) form a two-dimensional problem for each fixed A; by our result, this reduces to a one-dimensional problem.
116
3.6.10. Example: Dirichlet boundary value problem. Let n be a bounded C2 ,a_
domain inJK1 (n>2, a E ]O,l[), and consider the following Dirichlet problem: ,0,u+jJ.u ]
u
=
in n ,
f(x,u,>..)
=
°
(21)
em ,
in
where p. is an eigenvalue for the problem ]
°
,0,u + jJU u
=
in n , (22)
°
in eln .
We suppose that f : fi xJR x/\. ... JR is of class C1 , and satisfies f(x,O,O) =
°
L. : X
N:
Vx
°
E
n
(23)
{U E C2 ,a(fi) iu(x)=O, VxEeln} and Z
Let X
]
elf elu(x,O,O)
-+
u
Z
XX/\' -+
Z
+>
L.u = ,0,u+ p.u
(u,>..)
]
]
+>
N(u,>..) (x)
f(x,u(x),>..).
Our problem takes the fOTIn (24)
We know from the results of section 2.3 that L. is a Fredholm operator with J zero index. n
Let G be the group of transformations S : JR
-+
n JR , which are compositions
of translations, rotations and reflexions, and which leave the domain n invariant: Sen) = n. Define a representation of Gover CO(fi) (and so also over X and Z) as follows (r(S)u)(x)
= ueS -1 x)
°-
_.
Vu E C (n), Vx E n, VS E
G.
(25)
AsslUJle also that
117
V(x,u, A) E ~ xJR x fe, VS E G .
f(Sx,u,A) = f(x,u,A)
This condition is in particular satisfied when f(x,u,A) = f(u,A) does not depend explicitly on x. Application of corollary 2 gives the following result Let f in (21) satisfy (23) and (26), and let Gj = {SEG
I r(S)u=u,
VUEker Lj } .
Then each solution (U,A) of (21), with lIul1 2 ,a and II Ali sufficiently small, satisfies u(Sx) = u(x)
Vx
i1
E
VS
E
G. J
A more precise result can be given for bifurcation from the lowest value wl of (22). 3.6.11. Theorem. Consider the boundary value problem
L'lu + wlu u
= f(X,U,A)
in rI
=0
in art
where f satisfies (23) and (26), and Wl is the lowest eigenvalue .£or (22). Then each solution of (28) with lIull 2 and IIAIl sufficiently small satisfies: ,a u(Sx) = u(x)
Vx E ~ , VS E G .
Proof. From the foregoing it follows that it is sufficient to show that G1
=
G. By theorem 2.3.31 we have dim kel' L1
=
1; let ker L1
= {au,!
I aEJR}.
By the same theorem, u 1 can be chosen such that u 1 ex) > 0, Vx E rI. Since r (S)(kel' L1) = ker L1 , for each S E G, we have Vs
G,
E
for some a(S) E JR \{O}. Suppose
Ia(S) I < 2
3
1; since G is compact (rl is bounded)
we may suppose that the sequence S,S ,S , ... converges to some So E G (in the operator topology). But then 118
r(So)u 1 = lim r(Sn)u 1 = lim (a(S))nu1 = n->= n->=
°,
contradicting the fact that r(SO) has an inverse. In case la(S) I> 1, then . laC S- 1) [ = Ia(S) 1- 1 < 1, again leading to a contradiction. So Ia(S) I = 1. Finally, also a(S) = -1 is impossible, since (r(S)u 1)(x)
= u,(S -1 x) >
We conclude that f(S)u 1
=
°
Vx E
~
•
u 1 ' for each S E G. This proves the theorem.
3.6.12. Example. As a particular example of problem (21), let
~ =
B
0
=
{XElR2 [ II xII 2 < 1} be the lmi t ball in JR2 • The eigenvalues can easily be found
by using polar coordinates and a Fourier expansion in the angle variable. One finds that each eigenvalue ]J must satisfy :
(30)
for some kEN, J k (x) being the k-th order Bessel function. For each k E :N, equation (30) has a sequence of solutions {]Jkm I mEN}, tending to +00 as m-+ oo • The eigenvalues ]JOm (mE N) are simple, and correspond to radially symmetric eigenfunctions JOC/POmr); ]JOO is the lowest eigenvalue. For k
~
1, ]Jkm is a
double eigenvalue, the corresponding eigenfunctions being linear combinations of
Assume now that f is radially symmetric; that means, we consider the Dirichlet problem : L1u + llkmU =
u =
fer ,u,;\)
in B (31)
a
in dB
with
f(r,O,O)
= df du(r,O,O)
lir E [O,l] •
(32)
119
The synnnetry group leaving B invariant is the group 0(2). Next theorem follows then from corollaries 2 and 7. 3.6.13. Theorem. Under the foregoing conditions, let (u,A) be a solution of (31) , with II ull 2 (i)
u
and I All sufficiently small. Then :
,a
= u(r)
( 1"1" ) u ((211)) ¢ l(P x
1n case k
= u (x )
=0
VX E B- ,p
" = 0 , 1 ,"', k - 1,1n
case k
~
1 .
Also, for each such solution there is a S E]R such that the function vex) u(¢(S)x) is symmetric around the xl-axis:
v(ax) = vex)
or
v(r,-¢)
= v(r,¢)
o
(33)
So, when solving (31) for small solutions, one can restrict attention to solutions satisfying (33). In a later chapter we will discuss a 3-dimensional version of problem (31). 3.6.14. Syrmnetry of the von Karmtm equations. Another class of problems for which similar resul ts can be proved are the buckling problems for plates considered in section 2.4, and described by the von Karman equations. We want first to investigate the synnnetry properties of the operators appearing in the abstract formulation of the problem. The approach is the same "for each of the different cases considered in section 2.4 (clamped plate, simply supported rect@lgular plate and simply supported cylindrical plate); so we will not make the distinction here. Let Sl
C
JR2 be the basic domain of the plate problem, and let S :]R2
be @lY distance preserving transformation leaving
~
invariant :
S(~)
JR2
+
= ~.
Since the elements of H, the basic Hilbert space, are continuous functions, we can define the representation (r(S)u)(x)
= u(S -1 x)
Vu E H, Vx E
Let u,v,¢ E H be sufficiently smooth, and w
120
?i
B(u,v), then
(34)
This implies :
J~[ r(s)u,r(S)vl¢
=
f~[U'Vl(S-lX)¢(X)dX J~[ u,v j(x)¢(Sx)dx
f~I'lW(X).I'l¢(SX)dX J~ I'lw(S -1 x) .I'l¢(x)dx (r(S)w,¢) .
We conclude that B(r(S)u,r(s)v)
r(S)B(u,v)
Vu,vEH.
(35)
Vw E H .
(36)
It follows also that ccr(S)w) = r(S)C(w)
If the function FO appearing in the definition (2.4.13) of the operator A satisfies : Vx E
~
(37)
(that means: the external forces have the same symmetry as the domain ~), then also: Ar(S)
r(S)A .
(38)
Finally, also the operator Q appearing in the equation for a cylindrical plate satisfies Q(r(s)w) = r(S)Q(w)
Vw E H •
(39)
121
3.6.15. The clamped plate. Consider the buckling problem for a clamped plate, which is also subject to a normal load; this load is supposed to 'be proportional to a small parameter v. The equation takes the form : (I - '\A)w
+
C(w) = vp
(40)
2 2 where p is a fixed element of H = WO ' (rn. We let G be the group of transformations S considered in the preceding subsection; suppose that r(s)p
=p
VS
E
G.
(41 )
Then the solutions of (40) have s)~etry properties similar to those of the solutions of the Dirichlet problem considered in example 10. More precisely: Let .\. be a characteristic value of A, and L. I-A.A. Let J
Gj
J
J
{SE G I u(Sx) =u(x), VxErl, VuEker Lj }
Then for IX-'\j I and Ivl sufficiently small, each sufficiently small solution w of (40) satisfies w(Sx) = w(x)
Vx E rl , VS E G. . J
(42)
For example, in the case of a circular plate we have essentially tl;e same result as in theorem 13. In particular, when solving (40) for a circular plate, we can restrict attention to solutions which are symmetric around the Xl-axis. 3.6.16. The rectangular plate. In case p = 0 (or v = 0) in (40), this equation has an additional synnnetry : all operators corrrrnute with -I; if (w,.\) is a solution, then the same is true for (-w,.\). This will double the number of symmetry operators, as compared with the group G cons ide red in the preceding subsection. As an example, consider the simply supported rectangular plate. Denoting the elements of the group G by their representation over H, we have for this case:
122
(43)
where
rxw(x,y) = we -x,y) r yw(x,y) = w(x,-y)
(44)
rxy = r xr y . The characteristic values Amn (m,n E N\{O}) are given by (2.4.51); they correspond to the eigenfunctions ¢mn given by (2.4.52). Denoting by Gmn the subgroup of G which leaves ¢mn invariant, we have : G
mn
{I,-rx ,-ry ,r xy } if m = even and n
even ;
{I ,-rx,r y,-r xy}
if m = even and n
odd
{I,rx,-ry,-rxy}
if m = odd and n
even
{I,rx,ry,rxy }
if m = odd and n
odd
Solutions (W,A) with Ilwll H and IA-Amnlsufficiently small will be iIwariant under the opera tors of Gmn ,under the condition that Amn is a simple characteristic value. 3.6.17. The cylindrical plate. In the case of a simply supported plate the equation is no longer invariant under the operator -I, the normal load is zero; this is a consequence of the appearmlce dratic operator Q in the equation. Suppose that the normal load has the symmetry of the cylinder then we can take G
{rea) ,r(a)rx ,r(a)r y ,r(a)rxy I aEJR}
cylindrical even when of the qua(see (41));
(45)
where r x ' ry and r xy are as before, and 123
crCa)w)(x,y) = wCx,y+a)
Va E]R •
The subgroups Gmn corresponding to the characteristic values Amn given in subsection 2.4.28 are: G
G
{rea) ,r(a)r x I aE]R}
Gm,n
21T ,rC-p)r 2IT \ -} { rc-p) n n x p = 0,1, ... ,n-l
if m = odd and n
Gm,n
{rc~TIp) \ p = 0,1, ... ,n-l}
even and n
m,O
if
m = odd ;
Gm,O
if m
=
even;
if m
~
~
1
1 .
Solutions CW,A,V) in a neighbourhood of (0,Anu1,0) will be invariant under the operators of G . Also, when n > -I, one can restrict attention to solumn tions satisfying r w = w; this follows from corollary 7. Y
3.6.18. Bibliographicalp.ote. For other applications of symmetry argmnents in bifurcation theory (e.g. for the Benard problem), one can see the recent lecture notes by Sattinger [265].
124
Perturbations of symmetric nonlinear equations 4.1. INTRODUCTION
Consider the problem of finding 21T-periodic solutions of the ordinary differential equation : ~ +
x =
g(x,~} +
where g(O,O) =
Wp(t) ,
~~(O,O) = 0, p :
(1)
lR
-r
lR is continuous and 21T-periodic, and W
is a small parameter. In In any particular examples of such problem the function pet) is also even: p(-t)
=
pet). Under such condition one usually re-
stricts attention to 21T-periodic solutions which are also even in t. (See e.g. Hale [77], Hale and Rodrigues [ 88]). There is a very practical reason for this restriction: using the results of section 3.4 it is easily seen that the bifurcation equations for this problem reduce from two scalar equations in the general case to only one scalar equation if one restricts to even solu hons . In case W = 0 the restriction to even solutions can be justified by corol-
lary 3.6.7
each solution of (1) with
after an appropriate phase shift. When applicable; indeed, equation (1) is for
w
0 becomes an even function of t
t
lJ lJ
0 the corollary is no longer
t
0 not invariant under arbitrary
phase shifts. Although the restriction to even hlnctions gives us information
on Mme, solutions, the question remains whether or not there are any other solutions. In this chapter we describe conditions which ensure that all sufficiently small solutions of (1) are even when
lJ
to.
In the particular case of the Duffing equation, where g(x,~)
=
AX - x
3
(2)
such result was proved by Hale and Rodrigues [88 J. The main property of (1) ~ed
in their proof can best be eA~ressed by the hypothesis (R) used in
section 3.6. Let G
=
0(2), with a representation over the space of continuous
Zn-periodic functions as given by (3.6.19) and (3.6.20); let G'
=
{¢(O),o}. 125
°
°
For ~ = the equation (1) is equivariant with respect to (G,r); for ~ f the equation is only equivariant with respect to the subgroup ot. This means that the perturbation destroys the symmetry of the unperturbed equation. For
°
~ = (1) reduces to the equation studied in subsection 3.6.9; it follows that (1) satisfies the hypothesis (R) of section 3.6. The difference with the situation of section 3.6 is that (1) is no longer equivariant with respect to G = 0(2) when jJ f 0. In section 2 we prove some abstract theorems which generalize the result of Hale and Rodrigues to a large class of nonlinear equations having symmetry properties similar to those of (1). Our presentation is an adaption of some earlier joint work of H.M. Rodrigues and the author [182]. In section 3 we discuss the particular case where the unperturbed equation has an 0(2)symmetry; we also consider applications to periodic solutions of forced conservative oscillatory equations such as (1), elliptic boundary value problems in the unit circle, and some buckling problems having rotational symmetry. In section 4 we briefly discuss the situation when the unperturbed equation has an 0(3)-symmetry.
4. 2. THE ABSTRACT RESULTS
4.2.1. 1ne hypotheses. Let X, Z, A and L be real Banach spaces, rl a neighbourhood of the origin in XxAxl: and M : rl ->- Z; we will consider the equation
M(X,A,O) =
°.
(1)
Remark that we have replaced the parameter space A used in the previous
°
ter by a product space Ax=' When ° = we call (1) the Uf1pe~tLULbed equation, v.hile the p~LULbed eQuctUon. corresponds to f 0. We make the following assumptions about M. (Hl) M is of class C2 , M(O,O,O) thesis (C) of section 3.1.
°and
°
L
DI1(O,O,0) satisfies the hypo-
(H2) There is a compact topological group G, a closed subgroup GO' and sentations r : G -r LCX) and f : G ->- L(Z), such that M is equivariant with respect to (G,r,f) when ° = 0, and with respect to (Go,r,f) when
°f 126
°:
Vs E G, V(X,A,O) E n
M(r(S)X,A,O) = r(S)M(X,A,O)
(2)
and M(r(s)x,A,a) = r(S)M(X,A,a)
Vs EGO' V(X,A,a) En.
(3)
As in chapter 3 it follows from (Hl) and (H2) that L is equivariant with
respect to (G,r,r), and that we can find equivariant projections P E L(X) and Q E L(Z) such that R(P) = ker Land ker Q = R(L). We w~{,use the follo(J wing subspaces of X and Z : XI = {xEX I r(s)x=x, VSEG} , Xo = {xEX I r(s)x=x, VSEGO} , (4)
ZI = {ZEZ
I r(s)z=z,
VsEG} , Zo = {ZEZ I r(s)z=z, VSEGO} ,
together with the corresponding projections : PIX =
fG r(s)xdm(s) , Poz = f
r(s)xdmo(s)
Vx EX
GO Ql z =
fG r(s)zdm(s) , Qoz = f
(5)
r(s)zdmo(s)
Vz
E
Z
GO In (5) m and mO are the Haar measures of G and GO respectively; one has
We will also use the mappings y : G + 1R and y .
G + 1R defined by
1
yes) = sup{llCI-po)r(s- )ullluEker Lnxo' lIuli =1} and
(7)
yes) = inf{U (I-QO)f(S)wlI
I WE R(Q) n ZO'
IIwll = 1}
Now we can formulate the remaining hypotheses : (H3) For each u E ker L there is some s E G such that r(s)u E ker Ln XO;
i.e. :
127
(H4) There is as>
° such that
Sy(s) ,,;; yes)
Vs E G •
Hypothesis (H3) is in fact the hypothesis (R) used in section 3.6; (H4) a technical hypothesis, which can be verified as soon as ker Land R(Q) determined. At the end of this section We will briefly return on these hypotheses.
lS
4.2.2. Theorem. Assume (H1), (H2) and (H3). Then there is a neighbourhood W of (0,0) in XxI\. such that, for each (X,A) E W,
M(X,A,O)
=
°
(10)
implies the existence of some s E G such that f(s)x
E
Xo
'
i.e. f(t}f(s)x
f(s}x
Vt
E
GO •
Proof. This is just a restatement of theorem 3.6.6.
0
The next theorem forms the main result of this chapter; it describes conditions which ensure that the solutions of the perturbed equation (1) have the same symmetry as the equation itself. 4.2.3. Theorem. Assume (H1), (H2), (I-l3) and (H4). Let S c Z\{O} be a compact subset such that inf lIoll- 1l1QD M(O,O,O}oll _ o(S} > oES
Let
128
0
°.
be the cone generated by S. Then one can find a neighbourhood W of (0,0) in Xxf\, and a neighbourhood
w of 0 in L:, such that for each solution (X,A,O) r5
I 0, we have x
E
0
Wx(w n Cs ) of (1), with
XO' i. e . :
r(s)x = x p.r
E
Vs
E
(14) .
GO .
a f. Let (x, A,O) be a solution of (1), sufficiently near to the origin,
such that the Liapunov-Schmidt reduction holds. Let u x = x *(U,A,O)
and
F (u , A,0)
=
Px; then we have :
o.
(15)
Applying the results of chapter 3 separately for the cases
0 and
0=
0
I 0,
we see that x * and F have the following symmetry properties : x*(r(S)U,A,O)
r(s)x *(U,A,O) , Vs
F(r(S)U,A,O)
E
(16)
GO if
0
f 0, Vs
E
G if
0
0
r(s)F(U,A,O) Vs
By (H3) we can find some s
E
(17) E
GO if
0
f 0, Vs
E
G if (}
Gand some Uo E ker L n
r(s)u O' where, for simplicity of notation, we put
=
0 .
Xo such that u =
5 = s -1 for each s
E
G.
Then (15) implies that
o. The mapping G is defined for all s
E
( 18)
G and for (UO,A,O) in a sufficiently
small neighbourhood of the origin in (ker L n XO) x f\ x L:. It follows from (17)
that
o
( 19)
o
(20)
and
129
Let us analyse G in somewhat more detail. Using (19) we can write .
1
G(S,UO,A,O) = (I-QO)r(s) foDoFCr(S)Uo,A,to).Odt 1
(I-Qo)f(s)IoDoF(Por(~)uo,A,tO).Odt
1 +
1
(I-Qo)f(s)fodtfodt'
DuDaF(POr(~)uO + t' (I-PO)r(s)uO,A,to). (0, (I-PO)r(s)u O)
We have (21 ) where C2 (S,U O,A,0) remains bounded for s E G and (uO,A,a) in a neighbourhood of the origin; indeed G is compact and Du DaF(u,A,a) is continuous. Since F(POU,A,O) E R(Q) n Zo for each (U,A,O), it follows that Di'(POU,A,O) .0' E R(Q) n ZO' Then the definition of yes) implies that (22) where, for
° i=
0 : (23)
Since DoF(u,A,O) is a continuous function of its arguments, and
we have by (12) and the compactness of S
for all s E G and all (UO,A,O) sufficiently near to the origin, with and a E CS . Now suppose the theorem is false. Then we can find 130
° =I
0
nEN}, converging to (0,0,0) as n
+
each term being a solution of (1),
00,
°
n E CS\{O} and xn ¢ XU' Let u n = Pxn ; then x n = x'(un ,An ,0n ) for n large enough. Since u n E Xo implies xn E XU' by (16), we have u ¢ XU' Let s E G and u o n n ,n E ker LnxO be such that u = res )u o ; then it follows from the definition of yes) and from n n ,n f for all n EN. Moreover, our foregoing analysis (I-POJun f 0 that y(s) n and such that for each n EN we have
°
shows that (y(sn )110n II)
-1
G(s n ,u O,n ,A n ,0) n
Taking the limit for n
+
00
°
=
Vn EN .
(25)
and using the estimates (21), (22) and the hypo-
thesis (I-l4) , we conclude that: lim C (s ,u ,A,O) = n+oo 1 n O,n n n
°.
(26)
o
This, however, contradicts (24).
4.2.4. Corollary. Assume (H1)-(I-l4). Suppose that dim L
operator QDoM(O,O,O) : L
+
0 such that for each p E Cw = {lJP 1 pE w,aER} with 0 < IIpll ,.;;; aO' each solution (x,A) E. W of (28) is such that x E XO' Proof. Let ~ = Zo and w a neighbourhood of Po in Zo on which I QPII remains bounded away from zero; such neighbourhood exists because of condition (iii). If dim ~ < 00 we can innnediately apply theorem 3, with S = w. In the general case one has to reconsider the last part of the proof of theorem 3; from the expression for C,(s,uO,A,a) it is easily seen that (26) gives a contradiction. Also, a careful examination of the proof shows that it is sufficient for M to be of class C'. 0 4.2.7. Corollary. Under the conditions of corollary 6, consider the equation M(x,A) = 0p0 ' wi th a E R. Then there is a neighbourhood W of the origin in XxA and a 132
(29)
°0 > 0 such that, that x
E
XO.
for 0 < D
lal
~ aO' each solution (x,A) E W of (29) is such
. 4.2.8. Remark. Let the hypotheses (H3) and (H4) be satisfied for some closed subgroup GO of G, and let sl E G. Consider the subgroup : (30)
We claim that (H3) and (H4) are also satisfied for the subgroup Gl • Indeed, we have : (31)
This shows that x E Xl if and only if x = r(sl)P Or(sl)x. So the corresponding projection on Xl is given by Pl = r(sl)P Or(sl). Now we have: ker L = {r(s)uO I sEG, uOEker LnxO} =
{r(s.sl)r(sl)uO I SEG, uOEker LnXO}
= {r(s)u l
I sEG,
u l Eker Lnx l }
This proves (H3) for the subgroup Gl • As for (H4) , we have for each s E G
and similarly
It follows that Sl Yl(s) ~ Yl(s) for each s E G, with S1
=
-
~
~
-
S[ I r (s 1) I I r (s 1) 1111 r (s 1) 1111 r (s 1) II]
-1
. 133
Using this remark one can easily prove some variants of the foregoing results. For example, one obtains the following modification of corollary 7. 4.Z.9. Corollary. In the hypotheses of corollary 7, replace the condition Po E Zo by the condition f(sl)PO E ZO' for some sl E G. Then the solution of (Z9) considered in the conclusion of corollary 7 will satisfy :
o
(31 )
4.Z.10. We conclude this section with a remark on the hypotheses (Hl)-(H4). The following situation frequently appears in applications : X is continuously imbedded and dense in Z, while Z = ker L ffi R(L), and dim ker L < Let Q be a projection in Z onto ker L, and such that ker Q = -R(L); then we can take P = Qi x ' Assume also that res) = r(s)i x ' for each s E G; we win denote both representations by f. The claim is that under such conditions the hypotheses (H1) - (H4) imply that the restriction of f to ker L is irreducible over ker L, except when ker L C XO' This last case is uninteresting as far as theorem 3 is concerned, since the conclusion of theorem 3 is in that case an immediate consequence of the results of chapter 3. In order to prove the claim, suppose that ker L Ul ffi UZ' with Ul and Uz nontrivial subspaces, such that 00.
Vs E G •
By (H3) we may assume that Ui nxO f {a}, for i = 1,Z. Excluding the case ker L C XO' we may also suppose that e. g. Uz\XO f if>. Take u l E Ul n XO' u l f 0, and Uz E Uz\ XO' By (H3) there is some s E G such that r(s)(u l +u Z) E XO' i.e. such that r(s)u l
E
Ul n Xo
and
Since u l E Xo and r(s)u l E Xl it follows that yes) and
134
O. Also, r(s)u z E Xo
imply that yes) > 0. This however contradicts (H4). 4.3. PERTURBATIONS OF EQUATIONS WITH 0(2)-SYMMETRY
In this section we apply the preceding results to the particular case where the unperturbed equation is equivariant with respect to the symmetry group G = 0(2). We will denote the elements of 0(2) by cp(a) and Tocp(a), where aEJRand: cp(a)
=[
cosa -S1-na
Sinal
(1)
cosa
LORn) is a representation, then we denote rCa) = r(cp(a)) and = r(T)r(cp(a)). We refer to section 2.6 for the irreducible T T representations of 0(2) . If r : 0(2)
r
(a) =
+
r .r(a)
4.3.1. The hypotheses. We will consider equations of the form
M(X,A,O)
=
°
(2)
with hypotheses on M similar to the ones described in subsection 4.2.10. ~lore precisely, we assume : (a) M : XxI\xL: + Z is of class C2 and M(O,O,O) imbedded and dense in Z; (b) if L = DxM(O,O,O), then dim ker L < (c) there is a representation r : 0(2)
M(r(R)x,A,O) = r(R)M(x,A,O)
00
+
0, where X is continuously
and Z = ker L ffi R(L); L(Z) such that
\IR E 0(2), V(X,A) E XxI\.
(3)
(d) the restriction of r to ker L is irreducible. By the theory of section 2.6 the hypothesis (d) implies that dim ker L = 1
or 2. We will consider separately the three possibilities given by theorem 2.6.19. 4.3.2. Theorem. Assume (a)-(d), dim ker L
1 and r u = u for each u E ker L. T
Suppose also that
135
,
,
M(r x,A,o) = r M(x,A,o)
V(x,A,a) .
(4)
Then each solution (x,A,o) of (2), sufficiently near to the origin, satisfies :
,
r x =x .
(5)
Proof. This follows from theorem 3.6.1, using the basic group GO {¢CO),,}.
0
4.3.3. Theorem. Assume (a)-(d), dim ker L Suppose (4) and
M(-x,A,o) = -M(x,A,o)
,
1 and f u
-u for each uE ker L.
V(x,A,o) .
(6)
Then each solution (x,A,o) of (2), sufficiently near to the origin, satisfies :
r ,x
-x.
Proof. Define
rea)
=
rca)
(7)
r
0(2) ~ L(Z) by
r, (a)
= -r , (a)
Va
Em..
(8)
r
It is immediate that defines a new representation of 0(2), while (3), (4) and (6) imply that the conditions of theorem 2 are satisfied, if we replace r by The result follows then from theorem 2. 0
r.
4.3.4. Lemma. Assume (a)-(d) and dim ker L = 2. Let GO = {m(O),,}. Then the hypotheses (H3) and (H4) of section 2 are satisfied. Proof. By theorem 2.6.19 there is a basis {u 1 ,u 2} of ker L such that f(a)u 1 = coska.u 1 - sinka.u 2
r(a)u 2 = sinka.u 1
136
+
coska.u 2 '
Va
Em.
(9)
for sane k E N\{O}. It follows that ker L () Xo = span{u 1}. Also, for each a,b E lR we can find some p ;;;. 0 and some a E lR such that a
= pcoska
= -psinka •
b
'Iben we have :
which proves (H3). 1 As for (H4), we have Po = QO = 2(I+r,). It follows that
'Ibis shows that: yea)
=
,
y (a)
=
yea)
=
Y, (a)
=
Isinkal
,
Va
E lR ,
and that the hypothesis (H4) is satisfied. 0 Using remark 2.9 we see that (H3) and (H4) will also be satisfied if we take GO = {~(0)"o~(2aO)}' for some fixed a O E lR. Then Xo consists of those x E X such that (10)
4.3.5. Theorem. Assume (a)-(d), dim ker L = 2 and : (i) Mis equivariant with respect to the subgroup GO (4) holds; (ii) for a compact S
C ~\{O}
{HO),,}, i.e.
we have
o(S) :: inf lIoll- 1l1QD M(O,O,O)oll aES a
> 0
(11)
Then there is a neighbourhood W of (0,0) in XXA, and a neighbourhood w of 0 in ~, such that each solution (x, A,o) E Wx (CS () w) of (2), with a f 0, satisfies (5). 137
Proof. This is an immediate consequence of theorem 2.3.
0
4.3.6. Corollary. If in theorem 2 and theorem 5 the condition (4) is replaced by r T (2a O)M(x,A,a)
(12)
V(x,A,a) E XXAXL for some a O E R, then the theorems remain valid, if we replace the conclusion (5) by (10). Proof. This follows from the remark before theorem 5, using GO
{¢(0),T o ¢(2aO)}'
0
4.3.7. Application: Periodic perturbations of autonomous oscillation equations. Consider the problem of finding 21T-periodic solutions of the scalar equation
x + x = g(X,A)
+ h(t,x,A,a)
(13)
We assume the following : (i)
g: RxA
-+
g(O,O)
=
R is of class C2 , with
o
0
(ii) h : R xR x A XL
-+
R is of class C2 , 21T-periodic in t, and
h(t,x,A,O) = 0
V(t,X,A) •
Using the formalism of subsection 3.6.9, this problem can be brought in the form (2). Using the representation (r(a)x)(t) = x(t+a)
(r T (a)x)(t) = xC-t+a) ,
the hypotheses (a)-Cd) are easily verified. We have ker L = span{cos(.),sin(.)} , 138
which transforms under r according to (9), with k we can take: 1
(Qz)(t) = -eost IT
f2IT
°
1
z(s)cossds +-sint IT
f2IT
°
1. For the projection
z(s)sinsds,
.
Q
(16)
Vz E Z
Application of theorem 5 gives the following results. 4.3.8. Theorem. Suppose that
h(-t,x,A,a)
= h(t,x,A,a)
Let S be a compact subset of 8(S)
=
(17)
~\{o}
such that
inf lIall- 1 1 f 2ITCOStD h(t,O,O,O)adtl >
°
aES
()
°.
(18)
Then for nil and II all sufficiently small, a E Cs and a f 0, each sufficiently small 2IT-periodic solution of (13) will be an even function of t : x(-t) = x(t)
Vt E lR •
(19)
Proof. (17) and (19) correspond to the conditions (i) and (ii) of theorem 5. 0 4.3.9. Particular case. Let
~
= lR and
h(t,x,A,a) = ap(t) ,
(20)
where pet) is continuous, 2IT-periodic and even. If
f
2IT
°
cost p(t)dt f
°,
(21)
then, for II All and 1a 1 sufficiently small, and for a f 0, each sufficiently small 2IT-periodic solution of
x+
x
= g(X,A)
+
ap(t)
(22) 139
will be an even function of t. For the case of the Duffing equation this sul t was proved by Hale and Rodrigues in [88]. Remark that for this case is sufficient for g(X,A) to be of class C1 (see corollary 2.6). 4.3.10. Theorem. Suppose that g( -X,A)
-g(X,A)
V(X,:A)
and h(-t,-x,A,o) = -h(t,x,A,o) ,
V(t,X,A,O)
(24)
Let S be a compact subset of Z\{o} such that 6(S) = inf DES
~o~-1\J2nBintD h(t,O,O,O)odt\ ° °
>
°.
(25)
Then, for IIA~ and Ilo~ sufficiently small, with ° E Cs and ° f 0, each sufficiently small 2n-periodic solution of (13) will be odd in t : X( -t)
=
VtEill..
-x(t)
(26)
Proof. This result is proved in a similar way as theorem 8, this time using the following representation of 0(2) over Z, the space of continuous 2n-periodic functions :
(fCex)z)(t)
z(t+ex)
(f
T
(ex) z)(t)
-z(-t+ex) .
o
(27)
4.3.11. Application: an elliptic boundary value problem. Let B = {XEill.2 \ ~x~ < 1} be the unit sphere in ill. 2 . (11.11 is the Euclidean nonn). We will denote by (r,e) the polar coordinates of a point x E B ; correspondingly, the value of a function u at the point x will be denoted by u(x) or u(r,e). Let f B xill. x l\ -+ ill. and h B xill. x l\ -+ ill. be C2-functions; we will assume that f does not depend on e :
f(x,u,A) = f(r,u,A) , 140
(28)
af f(x,O,O) = au(x,O,O) =
°
Vx E
B
(29)
Consider the following boundary value problem !'Iu + flkmU u(x) =
f(x,u,A) + oh(x,u,A)
xEB
°
X E
(30)
aB
Here 0 E R is a small parameter, and flkm is an eigenvalue for the Dirichlet problem for the Laplacian in B !'Iu + flU
=
u(x) =
°
°
xEB, X E
(31)
aB .
We have seen in subsection 3.6.10 how the problem (30) can be brought into 2 a (B) - and Z = C ' a (B) - for some a E ] 0, 1[ • the abstract fonn (2). We have X = CO' The operator L : X + Z defined by
°
(32) is a Fredholm operator with zero index (see section 2.3). The representation r of 0(2) over CO(B), as introduced in section 3.6, takes the form : (r(a)u)(r,8) = u(r,8+a)
(r (a)u)(r,8) = u(r,-8+a) . T
.
(33)
The discussion of the solutions of (31), as given in section 2.3 and subsection 3.6.12, shows that the hypothesis (a)-(d) of the foregoing general theory are satisfied. We will denote the eigenfunctions, given in 3.6.12, as follows :
Xo ,m(r,8)
=
°
J (lflOmr)
k
=
Xk ,m (r,8) = Jk(lflkmr)cosk8
k;;;.
1:k ,m(r,8) = Jk (lflkmr )sink8
k;;;.
° (34)
141
For the projection Q we take : (Qu)(x)
=
~Xkm(x)
fB u(y)Xkm(y)dy
+ '1ooskm(x) fB u(y) skm(y)dy Here if k
'100 is a normalisation constant, such that Q is indeed a projection; = 0, then the second term in (35) does not appear.
4.3.12. Theorem. Let k h(r,-6,u,A)
=
=
° in (30). Assume that
h(r,e,u,A)
V(r,e,u,A)
Then each solution of (30), with lIull 2,a ,II All and satisfies : u(r,-e) = u(r,e) Proof. If k
=
lal
(36) sufficiently small,
VCr ,e) .
0, then dim ker L
4.3.13. Theorem. Let k
~
=
(37)
1, and theorem 2 applies.
0
1 in (30). Assume (36) and
fB h(x,O,O)Xkm(x)dx f
°
Then each solution of (30), with a f small, satisfies (37).
(38)
° and I ull 2,a ,II All
Proof. By application of theorem 5.
and Ia I sufficiently
0
4.3.14. Corollary. If in theorem 12 or theorem 13 the condition (36) is replaced by h(r,eo-e,u,A)
=
h(r,eO+e,u,A)
V(r,e,u,A)
(39)
for some eO E lR, then the theorems remain valid, i f in the conclusion (37) is replaced by 142
VCr ,e)
(40)
In the case of theorem 13, the condition (38) must also be replaced by (41)
pro a f. By application of corollary 6. 4.3.15. Theorem. Let k
~
0
1 in (30), and assume that
(i)
f(r,-u,A) = -f(r,u,A)
V(r,u,A) ;
(42)
(ii)
h(r,-e,-u,A) = -h(r,e,u,A)
V(r,e,u,A)
(43)
(iii)
IB
h(x,O,O)skm(x)dx
Then each solution of (30), with and a 0, will be such that
r
u(r,-e) = -u(r,e)
r °.
(44)
lIull 2,a ' HAll and icri sufficiently small,
(45)
V(r,e)
Proof. The proof is similar to that of the foregoing theorems, by using this time the following representation of 0(2) over COCB) (r(a)u) (r,e) = u(r,e+a)
(r (a)u) = -u(r,-e+a) T
o
(46)
4.3.16. Remark. Theorem 15 has a corollary analogous to the corollary 14 of theorem 13. Also, one can combine several of the foregoing results, on condition that the function h has enough symmetry. The main point to be careful about is that the symmetry imposed on h does not contradict the condition (41). We give a few examples. 4.3.17. Corollary. Let k be even in (30). Suppose that: (i)
h(-x 1,x 2 ,u,A) = h(x1 ,-X 2 ,U,A) = h(x 1 ,x 2 ,u,A) ,
(47)
for all values of the arguments (ii) (38) holds, in case k
r O. 143
Then each solution of (30), with lIull 2 ,11'\11 and ,a with 0 f 0, will be such that :
101
sufficiently small,
Proof. By combination of theorem 12 (if k = 0) or theorem 13 (if k f 0) with corollary 14, taking 80 = TI/2. One could also use theorem 2.3 directly, by taking GO = {4J(o),4J(TI),LO¢(O), LO¢(TI)}. If k is odd, then the condition (47) on h implies that both integrals in (41) vanish; so the conclusion only holds for k even. 4.3.18.
Coroll~ry.
0
Let k be odd in (30). Assume (42), (44) and ( 49)
Then each solution of (30), with lIuli Z ,11'\11 and ,a wi th 0 f 0, will be such that
101
sufficiently small, and
(50) Proof. By combination of theorem 15 with corollary 14, in which one takes 80 = TI/2. Again, under the sY1lllnetry condition (49) for h, (41) can only be satisfied if k is odd.
0
4.3.19. Remark. For k > 1, the elements of ker L remain invariant under the symmetry operators :
{r(~j)
1
j
= 0,1, ... ,k-1}
It follows from the theory of chapter 3 that, if
her,8 + iTIj,u,'\) = h(r,8,u,;.)
j
0,1, ... ,k-1 ,
j
O,l, ... ,k-l ,
then also 2TI .) =ur,8 ( ) ( ur,8+TJ 144
for sufficiently small solutions of (30). In case k group of 0(2) to get a similar result.
0 one can use any sub-
4.3.20. Application: the circular and cylindrical plate. The buckling problems for a clamped circular plate and for a simply supported cylindrical plate have, in the absence of normals loads, also an 0(2)-symmetry; in the case of the cylindrical plate the symmetry group is even larger than 0(2) . A physically interesting perturbation is obtained by the introduction of a normal load. The problem dealt with in this chapter reduces for these examples to the question under wllat conditions the symmetry of the normal load determines the symmetry of the corresponding solution. Our abstract results can be applied in a way similar as for the problem (30). Here we will simply state a few of the results which one can obtain. Using the appropriate Hilbert space H (see section 2.4) the buckling problem for the clamped circular plate is described by the equation : (I-AA)w + C(w) = vp
(53)
here v is a small parameter, controlling the amplitude of the normal load. For the simply supported cylindrical plate we have the equation : 2 2 (I-AA+a A )w + aQ(w) + C(w) = vp •
(54)
For problem (53) we denote by A (m = 1,2, ... , n = 0,1,2, ... ) the charactenm ristic values of A; for n = 0, AmO corresponds to a radially symmetric eigenfunction: (55) for n ;;;, 1, Anm corresponds to the eigenfunctions a
nm
(r)cosne
(56)
(r)sinne.
(57)
and a
nm
145
In the case of the cylindrical plate Amn will denote the characteristic values given in subsection 2.4.28, with corresponding eigenfunctions ¢ and mn Wmn , as given in (2.8.71)-(2.8.73). If we assume that all characteristic values are distinct, then the hypotheses (a)-(d) of this section can easily be verified, using the symmetry operators introduced in section 3.6, and using the fact that A is self-adjoint. 4.3.21. Theorem. Assume that
V(r,e) for some
eO
E JR. Let mE
IN\{O}
(58)
and n E IN; in case n;;'
1,
assume that
((p'¢mn)'(p,Wmn )) f (0,0) .
(59)
Then each solution (W,A,V) of (53), with IIwll, [A-Amn [ and [v[ sufficiently small, and with v f 0 in case n;;' 1, will be such that VCr,e).
o
(60)
4.3.22. Theorem. If the condition (58) of theorem 21 is replaced by
V(r,e) ,
(61 )
then in the conclusion (60) should be replaced by V(r,e) . 4.3.23. Theorem. Consider (54) and
aSSlli~e
o
(62)
that (63)
V(x,y) ,
for some yo E JR. Let mE IN\{O} and n E IN. In case n;;' 1, let (59) be satisfied. Then each solution of (54), with IIwll, [A-Amn [ and [v[ sufficiently small, and wi th v f 0 if n ;;. 1, will be such that V(x,y) . 146
o
(64)
4.3.24. Remark. ·For the cylindrical plate there is no analogue of theorem~ 22; the reason for this is the presence of the quadratic term Q(~). 4.4. AXISYMMETRIC PER1URBATIONS OF A PROBLEM WIlli 0(3)-SYMMETRY In this section we briefly discuss an application 6f theorem 2.3 and its corollaries to perturbations of a boundary value problem with 0(3)-symmetry. The results which we can obtain are far less general than for perturbations of problems with 0(2)-symmetries. This is mainly due to the fact that the irreducible representations of 0(3) (and, more generally of O(n) with n ~ 3) are much more complicated than those of 0(2). The main difficulty is in finding a representation of 0(3) and a subgroup GO for which (H3) and (H4) are satisfied. We will only give one particular example. 4.4.1. The problem. In dary value problem : ~u + ~£jU
u(x)
=
B
= {XER3 I IIxll < 1} we consider the following boun-
= f(x,u,A)
+
xEB
crh(x,u,A)
(1)
°,
X E
Here f : B xR x A -+ R and h f(Rx,u,A)
3B
B xR x A -+ R are of class C2 , and such that
f(x,u,A)
VR
E
0(3)
(2)
Vx
E
B
(3)
and
3f f(x,O,O) = 3u (x,O,O) =
°
further cr E R is a small parameter, while Dirichlet problem ~u + ~u
u =
°
° in
~£j
is an eigenvalue for the
B
(4)
on 3B
(See further on for the notation ~£.). We bring this problem in the fo~ (2.1) by defining X = C~,a(B), Z = CO,a(B) and
147
M(U,A,O) (x)
llU(x) + ]J9, j u(x) - f(x,u(x) ,A) - oh(x,u(x) ,A) , Vx E
B.
(5)
The operator L = Du M(O,O,O) is given by : Lu(x) = L'lu(x) + )J9, j u(x)
Yu EX, Vx E :8
(6)
We know from the theory of section 2.3 that L is a Fredholm with index zero. On CO (:8) we can define the following representation of 0(3) f(R)u(x) = U(R- 1x)
Vx E :8 , VR E 0(3) .
(7)
Then it is clear from (2) that M(U,A,O) is equivariant with respect to (0(3),r) . 4.4.2. The eigenvalue problem (4). In order to obtain some more information on ker L, let us consider the eigenvalue problem (4). It appears that because of the spherical symmetry of this problem, it can be handled in a sui table way by using the spherical harmonics introduced in subsection 2.6.30. (See e.g. Courant and Hilbert [48]). The space U9, of spherical harmonics of order 9, has dimension 2S',+1; let {Y9,m I m= 1,2, ... ,2t+1} be a basis of US'" which is orthonormal with respect to the L,(S2)-inner product. One can show that ~ 2 {Y S', EN, m= 1 ,2, ... ,29,+1} fonus a complete orthonormal subset o,f L2 (S ). Then we can solve (4) by expanding u in spherical harmonics; one writes
J
u(r,8)
(8)
(where (r,8) are polar coordinates for x E:8) and brings (8) into (4). It follows that n9,mCr) must be a solution of DS',(]J)n = 0 , nCr) regular at r =
° and n(1) = ° ,
(9)
where D9,(]J) is a linear second order ordinary differential operator, singular at r = 0, and depends on 9, and ]J, but not on m. For fixed 9, problem (9) has nontrivial solutions if and only if ]J belongs to an infinite {]J 9,j I j EN} of eigenvalues; if ]J = ]J 9,j then (9) has a one-dimensional space 148
of solutions, spanned by a fllilction nQ,j (r). The eigenvalues llQ,j are strictly positive, llQ,j ~ 00 as j ~ 00, and llQ,j f llQ,fjf if Q, f Q,f. We conclude that (4) has nontrivial solutions if and only if II = llQ,j for some (£,j) E ~xN; to each eigenvalue ll£j there corresponds a (2Q,+1)-dimensional eigenspace, spanned by the functions m = 1,2, ... ,2£+1 . Under rotations the eigenspace transforms according to the irreducible representation r(£) of 0(3), introduced in subsection 2.6.30. So the index £ in iJ£j refers to the dimension 2Q,+1 of the corresponding eigenspace and to the way the eigenvectors transform llilder rotations. 4.4.3. The case Q, = O. First suppose that £ = 0 in (1). This means that dim ker L = 1 and that the elements of ker L are spherically symmetric. Using the theory of chapter 3 one then immediately proves the following : if GO is any closed subgroup of 0(3), and h(Rx,u,iI)
h(x,u,iI)
(9)
then each sufficiently small solution of (1) will satisfy u(Rx)
u(x)
( 10)
In particular an analogue of theorem 3.12 holds.
For a particular subgroup, namely GO = {I ,-I}, this result can be extended for all £ EN. Indeed, for general ,~ E ~ one has : Yu E ker L ,
(11 )
(see remark 2.6.32). Then the theory of chapter 3 gives us the following result. 4.4.4. Theorem. Suppose that £ is even in (1), and that h(-x,u,iI) = h(x,u,iI)
Y(x,u,iI) •
(12) 149
Then each sufficiently small solution (u,A,o) of (1) will be even u( -x)
=
u(x)
Vx E
Ii
( 13)
In case £ is odd, f(x, -u, A)
-f(x,u,A)
V(x,U, A)
(14 )
V(X,U,A) ,
(15)
and h(-x,-u,A) = -h(x,u,A)
then each sufficiently small solution will be odd u( -x)
-u(x)
Vx E
Ii
o
(16)
4.4.5. The case £ = 1. Let now £ = 1 in (1). It is immediately seen from the defini tion that to each spherical harmonic Y1 E U1 there corresponds a unique a E 1R3 such that
a.e
ve
E
S2· .
( 17)
0(3) ,
( 18)
Then
(r(R)Y 1)(e) = (Ra).e
VR E
and for each Y1 E U1 we can find some R E 0(3) such that (19)
where cElli and e 3 is the unit vector along the x3-axis. It follows that the hypothesis em) of section 2 is satisfied if we take (20) As for (H4) , we are in a si tua tion as described in subsection 2. 10. As a norm for elements in ker L = R(Q) we can take 1I all, the Euclidean norm of the
150
vector a ER3 which determines Y1 via (17). The restriction of ~ to ker L = R(Q) corresponds to projection of a onto the x3-axis. Using this correspondence it is easy to see that y(R) = y(R) for all R E 0(3). So also (H4) is satisfied, and we can apply the results of section 2.
4.4.6. Theorem. Let £ =
in (1). Suppose that
h(Rx,u,A) = h(x,u,A)
VR E GO '
(21)
and
J1
°
r2dr
J 2 h(r,6,0,0)n 1 · (r)6 3d6 f S
°.
Then, for 1 All and ICJ I sufficiently small and solution of (1) will satisfy : u(Rx) = u(x)
(22)
J CJ
f 0, each sufficiently small
VR E GO .
(23)
Proof. The result follows from corollary 2.5.The condition (22) implies Qh(.,O,O) f 0. Remark that, because of (21), the integral in (22) becomes zero if we replace 63 by 61 or 62 , 0
4.4.7. Remark. The vector en in the definition of GO can of course be replaced by any e E S2. We conclude that in general an axisymmetric perturbation will lead to axisymmetric solutions with the same axis, if £ = 1.
151
5 Generic bifurcation and symmetry
5.1. INTRODUCTION In this chapter we start the study of the bifurcation equation and the bifurcation set for the important case that dim ker L = codim RCL) = 1. It appears that the behaviour of the bifurcation set is mainly determined by the degree of the first nonvanishing parameter-independent term in the Taylor expansion of the bifurcation function. We will discuss the cases where the dominant term is either quadratic or cubic. Our presentation combines elements from Chow, Hale and Mallet-Paret [39] with the approach used in [229]. We will show that in both cases considered in this chapter it is possible to describe the bifurcation set as a finite union of submanifolds in the par~ meter space; each of these submanifolds has a finite codimension. For each value of the parameter in the same connected component of the complement of the bifurcation set, the bifurcation problem has the same number of solutions; This number of solutions can only change when A crosses the bifurcation set, i.e. when a bifurcation takes place. In section 2 we discuss the case where the dominant term is quadratic, while the case of a cubic dominant term is studied in sections 3 and 4. It appears that under a certain "generic condition" the bifurcation second case is cusp-shaped. In section 5 we analyse the situation when the equivariance of the equation under a symmetry group prevents the generic condition from being satisfied. An example of such a nongeneric situation can be obtained by considering the buckling problem for rectangular plates, subjected to symmetric normal loads; in section 6 we study how different types of symmetry for the normal load affect the corresponding bifurcation set. A number of results in this chapter show a close relationship to some results from singularity theory. We will briefly discuss this relationship. In this connection let us remark that we obtain our results by "classical" methods (rescaling techniques and the implicit function theorem), while singularity theory uses as one of its most important more difficult preparation theorem of Malgrange and Mather (see e.g. [26], [74]).
152
BIFURCATION PROBLEMS WITH A QUADRATIC DOMINANT TERM 5.2.1. The problem. Let X, Z and A be real Banach spaces, ~ C X and we A open subsets, and M : ~xw -+ Z a Cr-function, with r:;;' 1. We want to study the solution set of the equation
M(X,A)
=
0
(1)
under the following hypothesis (H1) For some (XO,A O) E ~ x w, we have M(XO,A O)
=
0, while L
DxM(xO,A O) is
a Fredholm operator, with dim ker L = codim R(L) = 1. More in particular, since simple examples (see chapter 1) show that (1) may have a different number of solutions for different values of the parameter, we would like to solve the following problem: determine a neighbourhood ~1 of Xo in X, a neighbourhood w1 of AO in A, and a partition of w1 ' such that for A varying within each of the subregions determined by the partition, the number of solutions x E ~1 of (1) remains constant. This will also determine the bifurcation set B for (1); this is the set of all A E w1 which are a bifurcation point for (1) at a solution x E ~1 (see the definition in chapter 1). Without loss of generality, we may assume that (0,0) E ~xw and (XO,A O)
=
(0,0).
5.2.2. TIle generic problem. A particular problem of the form just described is the following. Let X, Z and ~ be as before; let Cr(~;Z) denote the Banach space of all r-times continuously differentiable functions m : ~ -+ Z satisfying
Imlr
=
sup{llm(x)1I
+
I Drn(x) I
+ ••• +
IIDrm(x)ll I XE~}
0 if a(t..) = 0 i f a(A) < 0
Proof. By (8) and the continuity of DZF(p,A) we can find p neighbourhood w1 of the origin in A such that :
and
DZ pF(p,A) > 0 D F(o,A) > 0 P
Vp
F(o,A) > 0
[-0,0]
VA
E
w, ,
D F(-o,A) < 0 , P
VA
E
w, ,
F(-o,A) > 0
VA
E
w1
E
°> 0 and a
Let A E w1• Then Dl(p,A) is a strictly increasing function of p E [-0,0], and has exactly one zero in the interior of this interval; let PO(A) be this zero. The function A * PO(A) is continuously differentiable, as follows from an application of the implicit function theorem on the equation DpF(p,A) = 0
(9)
Also POCO) = O. It follows that the function p * F(p,A) has a strict minimum at the point 155
P = PO(A). Let a(A) be the corresponding minimal value (10)
TI1e equation (5) has no solution P E [-8,8] if a(A) > O. TI1ere is exactly one such solution if a(A) = 0, namely p = PO(A). Finally, (5) has two different solutions P E [-8,8] in case a(A) < 0. TI1is proves the theorem. 0
5.2.6. TI1eorem. Suppose (Hl) and (H2). TI1en there is a 0 > 0, a neighbourhood w1 of AO in A, mld a constant C > such that each solution (p,A) E [-8,o]xw1 of (5) satisfies
°
(11)
Proof. If the theorem is not true, then there is a sequence {(p ,A ) I n n nEN}, converging to (0,0), such that F(p,A) = 0, P f PO(A), while also -2 n n n n a(An).(Pn-PO(An )) converges to zero. By the definition of PO(A) and a(A) we have:
Dividing F(pn ,An ) = find
°by (pn-PO(An
TI1is, however, contradicts (8).
))2 and taking the limit for n
+
00, we
0
5.2.7. Corollary. Suppose (Hl) and (H2). Suppose also that there is a ~ E A such that (12)
TI1en A E w1 is a bifurcation point for (1) corresponding to a solution x E Ql' if and only if a(A) = 0. TI1e bifurcation set
156
{A E W1 I a(A) =o}
(13)
contains AO' and is a submanifold of A, with codimension equal to 1. proof. To see that (13) is a submanifold of codimension 1, we remark that DI, a(O).~ = QOD\M(O,O).~ !\.
VA EA.
Because of (12) we can find some '~ E A for which this expression is different from zero. So a : w1 -'>- IR is a submersion in a neighbourhood of AO; this proves the las t part of the theorem (see Lang [ 141 ]) . It is clear that bifurcation points of (1) corresponds to bifurcation points of (5), at least if we restrict to a small neighbourhood of the origin. When a(A) > 0 then (5) has no solution, and A cannot be a bifurcation point. When a(A) < 0 and F(p,A) = 0, then it follows from the proof of theorem 5 that DpF(p,A) f 0; an application of the implicit function theorem shows that A is not a bifurcation point. Let finally A E w1 be such that a(A) = O. It follows from (12) that each neighbourhood of A contains parameter values A' for which aCA') < 0; for such A' the equation F(p,A') = 0 has two different solutions. When A' -'>- ,\ in the region {A' E w1 I a(A') < O}, then the two conesponding solutions will converge to PO(A), as follows from theorem 6; also PO(A) is the unique solution of F(p,A) = O. 1bis shows that A is a bifurcation point. D
S.2.S. Remarks. TIle submanifold (13) divides the neighbourhood in two connected compon~nts w~+) = {A E w1 I a(A) >O} and w~-) = {A E w1 I a(A) .) ,>.) . 5.2.9. Theorem. Suppose (H1) is satisfied at the origin. Then there is a neighbourhood Q1xw1 of the origin in XxA and a 6 > 0 such that the only points (X,A) E Q1xw1 at which (Hl) is satisfied have the form (puO+ V' (puO' A) ,>.) for some (p, >.) E ] -6,0[ x w1 such that : F(p,A) = 0 and Dp F(p,A) = 0
(16 )
For all other (X,A) E Q1xw1 we have either M(X,A) f 0 or dim ker DxM(x,A) codim R(DxM(x,A)) = o. Proof. Since (H1) includes the condition M(x,>.) = 0, it is clear that (x,>.) should have the given form, with (p,>.) a solution of F(p,A) = 0; this is a consequence of the Liapunov-Schmidt reduction. For each (p,>.) near the origin we define
158
we have L(O,O) = L, and we want to determine ker L(p,A) and R(L(p,A)). To do . so, we consider the equation (17)
wi.th z E Z given. Let into two equations :
x
=
puO+v, with P EJR and
vE
ker P; then (17) splits
( 18)
(I -Q) z
and
(19) Moreover, differentiating the equation defining
V*
(pUO,A) , we find (20)
Multiplying this relation by
p,
and subtracting from (18), we obtain (21)
Since (I-Q)L(O,O) = L is an isomorphism between ker P and R(L) = ker Q, the same holds for (I-Q)L(p,A); denote by K(p,A) the inverse of this isomorphism. Then (21) gives : (22)
Substitution of (22) into (19) gives : (23)
This equation has a unique solution p for each z E Z if and only if DpF(p,A) f 0; then ker L(p,A) = {O} and R(L(p,A)) = Z. If DpF(p,A) = 0, then (24)
159
to see this, it is sufficient to put z : 0 in the preceding calculations. Under the senne condition (23) has a solution if and only if the right-hand side of (Z3) is zero. For (p,A) : (0,0), this right-hand side reduces to QO(z); since QO is a nontrivial functional, we conclude that codim R(L(p,A)) : 1 if D F(p,A) = O. This proves the theorem. 0 p
5.Z.10. Theorem. Suppose (Hl) and (HZ) are satisfied at the point (O,O)EXxA. Then the set of points (X,A) E S61xw1 at which (Hl) is satisfied is given by (Z5) At these points, also (HZ) is satisfied. Proof. The first part follows immediately from theorem 9 and the definition of alA). Let L1 (A) = L(PO(A),A), K1 (A): K(PO(A),A),
(see (Z4)), and
If alA) : 0, then Z1(A)
~
R(L 1 (A)) if and only if (Z6)
The left-hand side of this inequality is contimlous in A, and reduces for A = 0 to QOD;M(O,O)(uo'u O), which is different from zero by assumption. We conclude that (Z6) will be satisfied for all A in a sufficiently small neighbourhood of the origin in ~. In particular, (HZ) will be satisfied at the points of (Z5). 0 5.3. BIFURCATION PROBLEMS WITH A CUBIC DOMINANT TERM In this section we discuss the number of solutions of the bifurcation equation (Z.5) in the case where DZF(O,O) ~ 0 and D3F(0,0) t O. This case has alp p ready been treated in the basic paper of Chow, Hale and Mallet-Paret [39] , 160
also in Vanderbauwhede [ 229] . In this last paper a rescaling technique used. The presentation here keeps somewhat the middle between these contributions. Our results allow in a nwnher of cases an easy calculation of . the approximate form of the bifurcation set. The approach seems to be particularly useful when many parameters appear in the problem; we will illustrate this by an example in section 6. 5•.3.1. The hypothesis. In this section we replace the hypotheses (H2) of the
preceding section by : (H3) r
~
3, and the bifurcation function F(p,A) given by (2.5) satisfies:
o
(1)
and (2)
The condition (1) can be reformulated as (3)
(see the previous section). As for (2), we obtain from a direct calculation the following expression for D3F(0,0) : p
D~F(O,O)
=
QOD~M(O,O).(uo'uo'uo) 2
2
- 3QODxM(0, 0) . (uO' K(I -Q) DxM(O ,0) . (uO,u O)) . We remark that again (H3) is a condition on the function mO(x)
(4)
=
M(X,A O),
and does not involve parameter values A f AO . Let (5)
We may suppose that a O > 0; in case a O < 0, it suffices to replace Zo by -zO in the definition of QO' The next lemma's describe the solutions of the 161
bifurcation equation F(p,>")
=
0 .
(6)
°
5.3.2. Lemma. Suppose (H1) and (H3). Then there is a > 0, a neighbourhood w1 of the origin in 1\, and a C1-map y 1 : w1 ->- ffi such that for each A E w1 satisfying Y1(>");;' 0 the equation (1) has a unique solution p E ]-rS,o[. Moreover, Y1(0) = O. Proof. Since F(p,O) = aop3 + 0(p3) it follows from continuity arguments that we can find rS > 0 and a neighbourhood w1 of the origin in 1\ such that 3 p
D F(p,>..)
> 0
Vp
D2F(8,A)
>0
D2F(-rS,>..)
D F(8,>..)
>0
D F(-o,>..)
p
P
E
p
P
V>"
E
w1 '
(7)
< 0
V>"
E
w1
,
(8)
> 0
V>..
E
lU
,
(9)
VA
E
w1
[-0,0]
1
and FCo,>..) > 0
F(-o,>..) < 0
(10)
From (7) and (8) it follows that, for each A E w1 , the equation
D2pF(p,>..) = 0
(11 )
has a unique solution p = PO(>..) E 1-0,0[. The implicit function theorem shows that Po : w1 ->- ]-0 ,or is continuously differentiable. Let now (12) Application of theorem 2.5 on the equation D F(p,>..) = 0 p
( 13)
shows that (13) has no solution p E j-o,8[ when Y1(>") > 0, just one solution p = PO(>..) when Y1(>") = 0, and two solutions in case y,(A) < O. Fix some 162
AE w1 • If Y1(A) > 0, then the map p * F(p,A) is strictly increasing; because of (10) the equation (13) has exactly one solution in ] -0,0[. If Y1 (A) =0 then F(p,A) has a strictly positive derivative, except at the point p = PO(A). thiS implies that F(p,A) is strictly increasing, and again has a unique solution p E ]-0,0[. This proves the lemma. 0 5.3.3. Lemma. Using the notation of lemma 2, define, for A E w1 ( 14)
If A E w1 and Y1(A) ~ 0, then A can only be a bifurcation point for (6) at a solution p E ] -o,o[ if YO(A) = Y1 (A) = O. proof. It follows from the implicit function theorem that A E w1 can only be a bifurcation point for (6) at a solution p if F(p,A) = 0 and D F(p,A) = o. We see from the proof of lemma 2 that the equation D F(p,A) p p = 0 has no solution if Y1(A) > o. If Y1(A) = 0 then the equation has exactly one solution, namely p = PO(A). This will also be a solution of F(p,A) = 0 if and only if YO(A) = o. This proves the lemma. 0 Remark that for all A E w1 satisfying YO(A) = Y1(A) = 0 we have (15)
5.3.4. Lemma. Suppose (H1) and (H3). For A E w1 , let (16)
Then there exist functions s+(A,n) and s_(A,n), defined and continuous for AE w1 and Inl sufficiently small, and continuously differentiable for n f 0, such that: (17)
(ii) if Y1(A)';;;; 0, then the solutionsof(13) in ]-o,o[ are given by : (18)
163
where n(A)
=
(-Y (A)) 1/2
( 19)
1
Proof. First, let us study the equation
o.
H(p,n,A)
(20)
By the argument of theorem 5.2.6 it is easily shown that solutions of (20) in a neighbourhood of the origin will satisfy Ipi ,,;;; Clnl for some constant C> o. Therefore we put = ns in (20), which gives us the equation:
p
(21)
The function Hl is of class C1 in all variables, and of class C2 in nand s; moreover H1(0,s,A) = DnH(O'S,A) = 0 for all (S,A). So we can write: (22) where HZ(n,s,A) = f01 sds
f1 ds'D 3F(PO(A) + SS'nS,A)S 2 - 1 .
o
P
(Z3)
" It follows from (ZZ) that in the domain n f 0 the function HZ has the same smoothness properties as the function H1. Further, H2 is continuous everywhere, and
H2 (0,s,A) = 3a(A)S Z - 1 ,
(Z4)
which shows that HZ(O,s,A) is continuously differentiable In s. Finally -7
lim n -D H1(n,s,A) n-+O s
=
6a(A)s ,
so that we can conclude that HZ has a continuous partial derivative in the variable s. Now Hz(0,±(3a(A))-1/ Z,A) 164
application of the implicit function theorem gives us the existence of functions s±(A,n) having the smoothness properties given in the statement of the lemma, satisfying (17) and such that for (n,A) near the origin, s = ,+(A,n) are the only solutions of H2 (n,s,A) = 0 in [-C,C]. Consequently the o~IY solutions of (20) in a neighbourhood of the origin are given by p = n,±(A,n). Part (ii) of the statement of the lemma then follows from the observation that (20) reduces to (13) if we put p = p-PO(A) and 0 n ~ (-Y1(A))1/2. Remark. It is possible to define the functions P+(A), as given by (18), for ~in a neighbourhood of the origin by taking-n(A) = 1Y1(A)11/2. The functions ob tained in this way are continuous, and of class C1 in {A E w1 I Y1 (A) f O}. MJreover, P± (A) are of class C2 in w = {A E w1 I Y1(A) < O}, since for A E w P = PiCA) solve (13), while F is of class C3 and D~F(P±(A)'A)
1
1,
f O. If we define yeA) for A E w1 by (25) then yeA) has the same smoothness properties as the functions PiCA), in particular, yeA) is of class C2 in w1 . 5.3.5. Lemma. Assume (H1) and CH3), and define yeA) by (25). Then we have the following for each A E W
1
(i)
if yeA) > 0, then (6) has exactly one simple solution pE]-o,o[;
(ii)
if yCA) = 0, then (6) has one simple and one double solution in ]-o,o[ ;
(iii) if yCA) < 0, then (6) has three simple solutions in ] -0, o[ . Proof. If Y1(A) < 0, then it is easily verified that F(p,A) has a local maximum at P = p_(A) and a local minimum at P = p+(A). It follows from (10) that (6) has one simple solution if the corresponding values of F(p,A) have the same sign, and three simple solutions when these values have an opposite sign. When one of these values is zero, then there is one simple and one 165
double solution. The result follows since yeA) is precisely the product of these maximum and minimum values of F(p,A). 0 5.3.6. 1nere is another way of characterizing the set {A E W1 I Y1(A) < 0 and yeA) =O}. It follows from the definition of YO(A) and Yl(,\) that (26) where R(p, A) is a continuous function which can be defined by 3 R(p,A) = f 0l s 2ds fl0 s'ds' fl0 ds"Dl(po(A) +sS'S"p,A)
(27)
It is then easily seen that, if Yl(A) < 0, then (28)
where n(A) = (-Yl(A)) 1/2 and (29)
The functions 0±(A,n) are continuous, and 0±(A,0) = ±2(27a(A))-1/2 .
(30)
The following theorem summarizes our results up to now. 5.3.7. Theorem. Assume (Hl) and (H3). Then there exist a 0 > 0, a neighbourhood w1 of the origin in fl., C1-functionals YO: w1 .. lR and Y1 : w1 .. lR, and continuous functionals 0+(A,n) and 0_(\,n), defined in a neighbourhood of the origin and satisfying (30), such that for A E w1 the following holds (i)
if Yl(\) > 0, then (6) has one simple solution p
E ]
-o,o[
(ii) if Yl(A) < 0, and if n(\) = (-Yl(A)) 1/2, then (6) has the following solutions in ] -o,o[ : (a) one simple solution if
166
(b) three simple solutions if
(c) one simple and one double solution if y,(A) < 0 and
(d) one triple solution if YO(A) = y,(A) = 0 .
5.3.8. Theorem. Assume (Hl) and (H3). Then
D
°> 0 and the neighbourhood w"
appearing in the statement of theorem 7, can be chosen sufficiently small such that each solution (p,A) E j-o,o[ x of (6) will satisfy:
w,
(3') for some constant C. Proof. If not, then we can find a sequence of solutions (Cpn,An ) I nEN} of (6), such that pn f PO(An ) for each n EN, and
as n + 00. Using the expression (26) for F(p,A), and dividing F(pn ,An ) = 0 by IPn-PO(An) 1 3 , we find in the limit for n + 00 : R(O,O) = 0 However, R(O,O) = aO' and so (32) contradicts (H3).
(32) D
We conclude this section by giving an alternative formulation of the hypothesis (H3). Since the definition of the function F(p,A) contains the projections P and Q, and the vectors uo and zO' we can ask whether (H3) will still be satisfied for another choice of these projections and vectors. Next '67
theorem shows that this is indeed the case. 5.3.9. Theorem. Suppose that mO(x) and Zo be as before, and define :
=
M(X,A O) satisfies (H1). Let P, Q, uo
(33) Suppose that r (i)
k
~
~
2. Then the following statements are equivalent
D~FO(O) = D~FO(O) = and
= Dkp- 1 F0 (0) = 0
,
D~FO(O) f 0 ;
(ii) there is a Ck-map X'" : j-o,o[ D X'" (0) f 0, such that
+
X, with X" (0).
=
Xo and
p
Vp E ] -0,0 [
(34)
for some zl ¢ R(DxMO(xO)) and some function a(p) satisfying a(O)
k = Dpa(O) = ... = Dk-1 p a(O) = 0 , Dpa(O). f 0 .
(35)
Proof. We remark first that (H1) implies that FO(O) = DpFO(O) = o. Assume (i), and let
Then (32) and (33) are satisfied, if we take zl = Zo and a(p) = FO(p). Conversely, suppose (ii) is satisfied. Let P(X'" (p)-x O) = S(p)u O; then S(O) = 0 and
(36)
for some function ~ : j-o,o[ (I -Q)mO(X" (p))
168
=
+
ker P. From (35),
a(p)(I -Q) zl '
and the equation defining V*(S(p)UO,A O) it follows that:
yeO) = a
DpyeO)
=
a ,
Dk - 1y(0) = p
a•
(37)
(38)
and (39) From (36) and (37) it follows that :
Indeed, the only term in D~F1(0) containing D~Y(O) has the form
which is zero by the definition of QO' Now QO(zl) f 0, since zl ~ R(DxmO(x O))' From F1(p) = a(p)~(zl) and (35) it follows that (40)
Finally, we compare the definition (39) of FZ(p) with the definition of FO(p). We obtain: FZCO) = FOCO) DpFZ(O) DZFZ(O) p
= =
DpFO.D pS(O) , DZFOCO). CD p(3(O))Z + Dp FZ(O) , p ' .DZS(O) p
DkFZ(O) = rfFOCO).CD p p pSCO))k
+
terms containing -1 0 (c) The solution set along YO(A) = 0
173
5.4.3. Remark. The condition of theorem 2 can be refonnulated as follows there exist ~1 E A and ~2 E. A such that
(7)
The derivatives appearing in (7) can be calculated as follows. It follows from the definition of YO(A) and Y1(A) (see (3.12) and (3.14)), from POCO) = 0 and from (H3) that VA E A
(8)
and (9)
Using the definition of F(p,A) one finds (10) To calculate DAY1(0).~ from (9), one needs DAv* (0,0); this can be obtained by differentiating the relation defining v*(U,A); one finds :
Finally one obtains
5.4.4. CorOllary. Consider the generic problem described in subsection 2.2. Suppose that mO E C3(~;Z) satisfies (H1) and (H3) at xO' Then the bifurcation set in a sufficiently small neighbourhood w1 of ~ in C3(~;Z) is given by :
174
pro a f. We only have to show that the condition (7) is satisfied (with A = C3 (Q;Z)). From (10) and (11) we find for this case
and
DmYl(~)·m = QODxm(xO)·u o - OoD~~(XO)·(UO,K(I-Q)m(Xo)) Vm
E
C3(Q;Z)
Now make the following choice for m1 and ~ : Vx E Q
and
The matrix in (7) becomes the identity matrix. This proves the corollary.
0
5.4.5. We can use corollary 4 to reformulate the condition of theorem 2. We suppose that not only M(x,A), but also DAM(x,A) is of class C3. We define the mapping, : we 11. -+ C3 (Q;Z) by ,(A)
=
M(.,A)
VA E w •
(13)
Let ~ = ,(A O)' and let;;; be a neighbourhood of ~ in C3 (Q;Z) such that for each mE;;; the map w(m) = (YO(m)'Yl(m)) is well-defined. We put So = {mE;;; I w(m) = o}; So is a C1-subrnanifold of C3 (Q;Z) with codimension equal to two.
Using this framework, the condition of theorem 2 can be restated as saying that the mapping Wo,: w1 = ,-1(;;;) -+:JR2 is submersive at the point AO• The proof of theorem 2 shows that this means that
where Tmo So is the tangent space to So a t ~. So the condition says nothing else than that the map, is transversal to So at A = AO. It is clear that 175
this can only be satisfied if dim A ? 2, since So has codimension 2. if dim A ? 2, then the transversality condition will be satisfied "generi-
cally', i.e. for almost all T : A ~ C3(~;Z). For this reason the transversality condition will also be called the geJ'lvUcc CCOYldi:UOYl, and the corresponding bifurcation set a geYlvUcc b~6LlJLc-atiOVl f.,e;t. 5.4.6. Sjmilarly as it was the case with the results of section 2, also the resul ts of sections 3 and 4 show a clear relationship with some resul ts from singularity theory; this relationship was already commented by Chow, Hale and Mallet-Paret in [39] and [40]. Suppose that M(X,A) is of class Coo, that dim A = P < and that M satisfies (H1) and (H3), say at (0,0). Then also the bifurcation function F(p,A) is of class Coo, and can be considered as an unfolding 'of the function FO(p) = F(p,O), which has the form FO(p) = aop3 + higher order terms. The transversality condition can be written as : 00
rank(a .. ) = 2
(i=1, ... ,p
1J
j=1,2),
(14 )
where, for i = 1, ... ,p :
of oYo ail = OA, (0,0) = OA. (0,0) 1
1
and
(15)
Assume that (14) is satisfied, then we may suppose, without loss of generality, that
This implies that the map (p,A 1 , ... ,Ap ) ~ (P,111, ... ,l1p ) given by p
=
p - Po (A)
111 = YO(A) 112 = Y1 (A) 11 1. A.1 176
(16) i
3, ... ,p
is a COO-diffeomorphism in a neighbourhood of the origin in lR xlRP • Then (3.26) shows that in the new variables (p,~) the bifurcation function takes the
( 17) with R(O,O) = aO. Comparing with the basic unfolding result of singularity theory this shows that the transversality condition (14) is the necessary and sufficient condition for F(p,A) to be a universal unfolding of FO(p). When this is the case, then there is a smooth change of coordinates near the origin of the form V·
1
=¢.(A) 1
i
1, ... ,p
P = r;(p,A)
such that in the variables (p,v), F(p,A) takes the normal form (18) The genericity of the condition (14) means the following. Consider the space of all Coo-mapping G : lR xlRP -+ lR. This space can be given the structure of a Baire topological space by introducing the so-called Whitney topology. Consider the subset U of all G(p,A) such that G(p,O) = FO(p); U is a linear submanifold. Then it follows from Thom's transversality theorem that the subset of all G E U which satisfy (14) is generic in U, i.e. forms an open and dense subset of U, equiped with the Whitney topology. (For the results on singulari ty theory, quoted in this subsection, we refer to Golubi tsky and Guillemin [72], Brocker [26] and Martinet [261 ]). 5.4.7. It is interesting to compare the bifurcation sets for the equations
° ° is given by
F1(P'~) = and F2 (p,v) = 0, with Fl and F2 given respectively by (17) and (18). It follows from our foregoing results that the bifurcation set for
F1(P'~)
(19)
177
with 0±(0,0) = ±2(27ao)-1/2. The bifurcation set for F2 (p,v) 62
=
{vE1RP
I v 2 ";; 0, 27V~ + 4v~ = O}
o is
given
,
as can easily be verified. By a rescaling of p one can achieve that a O = 1. 'Then one obtains (20) from (19) by replacing 0±(~) by its value at ~ = 0; similarly, one obtains (18) from (17) by replacing the remainder term R(p,~) by its value at (0,0). 5.4.8. We have obtained our bifurcation results for the equation (1) in a general setting, under relatively weak smoothness conditions, and using only classical tools such as rescaling techniques and the implicit function theorem. To the contrary, the approach via singularity theory requires that the equation is COO and that the nwnber of parameters is finite, while the proof uses the less elementary preparation theorem. (In a recent contribution, Michor [ 164] has extended the preparation theorem to the case where the parameter belongs to a Banach space). However, in more complicated situations the unified treatment via singularity theory seems to have some advantage; indeed, once the general theory is established, the application of the results of singularity theory to particular examples reduces to merely algebraic manipulations. For example, one could try to use our method to study bifurcation in the general case that the binlrcation function satisfies : F(O,O) = DpF(O,O) = ... = D~-1F(0,0) = 0 However, the treatment becomes very complicated for k > 3. Using singularity theory, the problem reduces to the study of the solutions of a polynomial equa tion of the form k
p
+
o,
with ai(A O) = O. For a somewhat different approach to bifurcation theory (in particular imperfect bifurcation) via singularity theory, see the basic papers by Golubitsky and Schaeffer ([ 74] and [ 75]).
178
.4.9. Non-generic bifurcation. We conclude this section with the description of a particular situation where the transversality condition (7) is not satisfied. This case is of some interest when the equation has additional symmetry, as we will see in the next section. The hypothesis is that yoO,) = a for all A E w1 ; Then the bifurcation equation has the solution P = PO(A) for each A E w1 . All other solutions near (0,0) can be written in the form (p,A) = (PO(A)+p,A), where Cp,A) has to be a solution of the equation 1
F1 CP,A)
=f
o
D F(pO(A)+sp,A)ds P
(21)
O.
Irrieed, F(PO(A)+p,A) pF 1 (p,A), and (21) is equivalent with F(PO(A)+p,A) = 0 if p f O. From (21) we see that : F1 (0,A) = Y1 CA) 1 2
a , (22)
LTDpF1(0,A) = a(A) ,
We can apply theorem 2.5 to equation (21); remembering that the bifurcation equation has always the solution P = PO(A), while (21) gives us the solutions p of PO(A), we obtain the following result. 5.4.10. Theorem. Suppose that the equation (1) satisfies the hypotheses (Hl) and (H3) at (0,0), while also YOCA) = a for all A E w1 . Assume that we can find some>: E A such that (23) Then the bifurcation set for Cl) is given by So = {A E w1 I YO(A) =Y1CA) =O} = {A E w1 I Yl(A) =O}, which is a C1-submanifold of A, with codimension 1. If Y1(A) > a then (1) has one simple solution in a sufficiently small neighOCurhood ~1 of the origin in X; this unique solution corresponds to the solution P = PO(A) of the bifurcation equation. If Y1(A) = a (i.e. A E SO) then the solution of (1) corresponding to P = POCA) remains the unique solution in ~1' but is now a triple solution. If Y1 CA) < 0, then (1) has three simple Solutions in ~1' one of which corresponds to POCA). 0 179
The following figures give a sketch of the bifurcation set and the solutions.
(a)
(b)
Fig. 6. (a) Bifurcation set (b) Solution set.
5.S. GENERIC CONDITIONS
h~
SYMMETRY
5.5.1. In this section we study how the symmetry of the equation M(X,A) = 0
(1)
can affect the bifurcation set. Our hypotheses are the same as in the preceding sections: we assume that M satisfies (H1) and (m) at (XO,A O) = (0,0). We will also assume that M is equivariant with respect to some (G,r,r), where G is a compact group. The projections P and Q used in the LiapunovSchmidt reduction of (1) are chosen such that they commute with the symmetry operators (see chapter 3). The bifurcation function
H(u,A) _ QM(U+V*(U,A) ,A) satisfies
180
U E
ker L,
AE
W
(2)
H(r(s)U,A) = r(S)H(U,A)
Vs E G , V(U,A)
(3)
The relation between H(U,A) and F(p,A) is given by V(p,A) .
(4)
5.5.2. The one-dimensional subspaces ker L = R(P) and R(Q) remain invariant under the operators res), respectively res). It follows that there exist scalar functions a : G -+ R and ~ : G -+ R such that : Vs
E
G
(5)
It is clear that both a and ~ are one-dimensional representations of G. Con-
sequently, {a(s) [SEG} is a compact subset ofR\{O}, which is closed under multiplication and inversion; the same holds for {~(s) [ s E G}. We conclude that
Vs
[a(s) [
E
G •
(6)
It follows from (3), (4) and (5) that FCa(s)p,A)ZO = H(a(s)puO,A) = H(r(S)pUO,A)
= r(s)H(PUO,A) = ~(S)F(p'A)ZO ' i.e.
F(a(s)p,A)
=
~(S)F(p,A)
Vs E G , V(p,A) .
(7)
5.5.3. Lemma. Let M be equivariant with respect to (G,r,r), and satisfy (H1) and (H3) at ( 0 , 0). Then :
a(s) = ~(s)
Vs
E
G •
(8)
Proof. It follows from (7) that 3 3
(a(s)) DpF(O,O)
=
~
3
a(s)D pF(O,O)
Vs
E
G •
181
Since Dp3p(0,0) f 0, we conclude that (a(s))3 = ;;:(s) , which together with (6) implies (8). 0 5.5.4. Define GO={SEG! a(s) =1} and assume that GO r(s)u = u
=
(9)
G, i.e. Vs E G , Vu E ker L .
(10)
Then it follows from theorem 3.6.1 that all solutions (x,A) of (1) in a neighbourhood of the origin in XxA will satisfy : r(s)x = x
Vs
G •
E
( 11)
The bifurcation problem can then be reformulated and discussed using the spaces Xo and Zo instead of X and Z (see chapter 3). In this reduced formulation symmetry plays no further role, and the general discussion of the preceding section applies. 5.5.5. Now suppose that GO is a proper subgroup of G, i.e. a(s) = -1 for some s E G. Then (6) implies that GO will be a no~al subgroup of G, i.e. s -1 DGODS = GO for all s E G. Application of theorem 3.6.1 shows that' all sufficiently small solutions (x,A) of (1) will satisfy r(s)x = x
Vs
E
GO .
( 12)
This allows us to restrict our attention to solutions (x,A) of (1), with x E XO' i.e. we can consider M as a mapping of the type M : XOXA + ZO' where
Xa
= {XEX! r(s)x=x, VSEGO} ,ZO = {ZEZ !r(s)z=z, VSEGO}.
(13)
The action of the group G on Xo and Zo reduces to the action of a two-element group G= {e,i}, with i 2 = e; the group Gis isomorphic to the quotient group G/GO. So the action of Gon Xo and Zo is given by : 182
r(e)x=x.
r(i)x
r(s)x
(14)
f(e)z = z
f(i)z = f(s)z
(15)
where s is an arbitrary element of G\G O' We have (16 )
and (7) shows that F(p,A) is odd in p : F(-p,A) = -F(p,A)
V(p,A)
( 17)
It follows that
(18)
So we have a case of non-generic bifurcation, as described at the end of the preceding section. 5.5.6. Theorem. Assume M satisfies (H1) and (H3), and is equivariant with respect to some (G,r,f), where G is a compact group. Define the subgroup GO by (9), and suppose the following: (i)
GO f G ;
(ii) there is some },
E 11.
such that
Then the bifurcation set for (1) is given by (19)
For all A E w1 the equation (1) has a solution V*(O,A) E XO' corresponding to the zero solution of the bifurcation equation. If y 1 (A) < 0 there are also two other solutions, which are invariant under the action of the subgroup GO' 183
and which can be obtained one from the other by application of res), sEG\GO these two solutions correspond to nontrivial solutions ±p of the bifurcation equation. 0 5.5.7. The proof of theorem 6 follows immediately from the foregoing remarks and theorem 4. 10. The condition (ii) corresponds to the transversali ty condition (4.23) of theorem 4.10. The following more direct proof, which uses the oddness of the bifurcation function F(p,A), was suggested to us by J.K. Hale. We have, from (17) : F1 (p,A) =
fo1 D F(op,A)do p
.
(20)
F1 (p,A) is of class C2 , and even in p. For p t- 0 the bifurcation equation F(p,A) = 0 is equivalent to :
o.
(21 )
Define G(p,A) by (22)
Since F1 (p,A) = Yl(A)+p2 R(p,A), with R(O,O) = aO' it is easily seen that G is of class C1 • Moreover, G(O,O) = 0 and D~G(O,O) = a O > O. Now F(p,A) = G(p2,A), and (p,A) is a soiution of (21) if and only if (p2,A) is a solution of G(p,A) = o. So we look for solutions of :
o
p
>
0 .
(23)
We can use the implicit function theorem to solve the equation in (23) : for each A near zero the equation G(p,A) = 0 has a unique solution p = peA). It is easily seen that this solution has the form : (24) where ~(A) is a CO-function, with ~(O) a~l (~ is of class C1 in the region Yl(A) f 0). It follows that ~(A) > 0 for all A near zero. 184
Now bifurcation occurs because of the condition
P~
0 in (23). We have
p(>J > 0 if Yl (A) < 0, and peA) < 0 if Yl (A» O. It follows that (21) has two nonzero solutions if Yl(A) < 0, and no such solutions if Yl(A) ~ O. The bifurcation points are given by Y1 (A) at the zero solution.
=
0, and the bifurcation takes place
5.5.8. Remark. There are more general situations than the one described in this section where the transversality condition of section 4 is not satisfied because of some symmetry properties of the operator M. To give an example, let mO = M(.,O), and suppose that mO is equivariant with respect to some (G,r,r). Under the hypotheses (Hl) and (H3) we can still determine the subgroup GO; suppose that GO f G. Suppose that A =JRP (as in section 4), and define mi = :~ (.,0). Finally, 1 suppose that for each i E {1,2, ... ,p} there is some si E G\G O such that:
res. )m. (x) 1
(25)
1
Then, using (4.15) and (4.10) we find
i
We conclude that ail = 0 for all i = 1, ... ,p,
1, ..• ,p .
the generic condition (4.14) will not be satisfied, although we do not necessarily have that YO(A) = 0 for all A. and
5.6. APPLICATION: 1HE VON KARMAN EQUATIONS FOR A RECTANGULAR PLATE 5.6.1. The problem. In this section we consider the buckling problem for a simply supported rectangular plate, subjected to a compressive thrust at its short edges and to a normal load, proportional to a small parameter v. In the appropriate Hilbert space H (see section 2.4 for the details) the equation for this problem takes the form : (I-AA)w
+
C(w) = vp ,
(1) 185
where p is some fixed element of H; the parameter A measures the magnitude of the compressive thrust. We want to Stlrly the bifurcation of solutions of (1) for (W,A,V) in a neighbourhood of (O,AO'O), where AO is a characteristic value of the operator A. These characteristic values were detennined in section 2.4, where we found
,i (m2+n ZQh 2
m,n
4.Q.2m2
1,2, . .. ,
(2)
wi th corresponding eigenflllctions
4.Q.3/2 . mn 7 2 2 slllzo (x+.Q.) n (m- +n .Q. )
¢mn = 2
nn
sin~
(y+1)
m,n
1 ,2, ...
(3)
Yv
Fix some (m,n) such that Am,n fA" m ,n for all (m' ,n') f (m,n). For notational convenience we denote Amn and ¢mn by AO and ¢O. Let (4)
Then ker L = span{¢O} and R(L) = {wEH I (w,¢O) = A}. For the projections P and Q used in the Liapunov-Schmidt reduction we take the orthogonal projection onto ker L : Vw
E
H .
(5)
5.6.2. Equivariance group when p = O. In subsection 3.6.16 we discussed already the symmetry of (1) when there is no normal load, i. e. when p = O. Each of the operators A, C and P COllllTlute with the operators of the finite group : (6)
where, as in (3.6.44) : (TXW)(X,y) = w(-x,y) Let
186
(TyW)(X,y) = w(x,-y)
Txy
T T
xy
.
(7)
(8)
this is precisely the subgroup GO defined by (5.9), here denoted by G1 for "notational convenience further on. This subgroup depends on our choice of (m,n). Since : Tefl x mn
=(-1)
m+l
ymn =(-1)
efl mn
Tefl
n+l
efl mn
(9)
it follows that (10)
where T1 = I , TZ = (-1)
m+l
(11)
TX' T3
Using this notation we have (lZ)
T·
J
j
1,Z,3,4
(13)
and TZT4 = T3 •
(14)
For general p E H equation (1) will only be equivariant with respect to the isotropy subgroup of p, that is the subgroup Gp
=
hE G I TP = p} .
We want to study how the bifurcation properties of (1) depend on the symmetry of p, i.e. how they depend on Gp . First we consider the irreducible representations of G1• 5.6.3. Irreducible representations of G1 • If T E G, then also -T E G; consequently, the action of G on an element wE H is uniquely determined by the 187
action of the subgroup Gl on w. For this reason we restrict our attention the subgroup Gl . Let r : Gl ~ LORn) be an irreducible representation of Gl . It follows (13) that for each, E Gl , r('l) has an eigenvalue equal to ±1. MOreover, the group Gl is commutative. Then corollary 2.6.10 implies that reT) = ±I each, E Gl . Since the representation is irreducible, it follows that n = Such an irreducible representation r : Gl ~JR = LOR) is uniquely determined by the real numbers (1., = r(,,) J
j = 1,2,3,4
J
moreover, (1.j = ±1 for each j, and (1.2(1.3 = (1.4' (1.3(1.4 = (1.2 and (1.2(1.4 = (1.3' These relations are sufficient to show explicitly that there are only four different irreducible representations of G, denoted by r i (i=1,2,3,4), and determined by the numbers
r, (,,) J
1
(1." 1J
E
i,j
JR
1,2,3,4
(15)
given in the following table
'1
'2
'3
'4
rl
1
1
1
1
r2
1
1
-1
-1
r3
1
-1
1
-1
r4
1
-1
-1
1
5.6.4. Remark 1
-4 1
It follows from the table above that
4
L (1.1'k(1.J'k
k=l
1J
4
4 i~l (1.ij(1.ik 188
1
=
-4
4
I . 1
Y
XCTJ·)Y(T J.)
x,y EX.
(20)
Now we define a representation r 0 : G1 ~ L(X) as follows 189
Vx EX, Vj,k = 1,2,3,4 .
(21)
It is clear that fO is orthogonal with respect to the inner product (20).
If f : G1 ~~ is an irremlcible representation, then we can consider f as an element of X, and Vj ,k
1,2,3,4 .
This means that (22) i.e. span{f} is an invariant subspace of X, whjch transforms tmder fO according to the irreducible representation f. Since by (19) different irreducible representations are orthogonal when considered as elements of X, it follows that G1 has at most four different irreducible representations. Let V be the subspace of X spanned by all the irreducible representations of G1 ; the foregoing shows that V is invariant under the representation fO' If V X, then also V 1 is invariant uIlder f O,and Vi contains a one-dimensional subspace span{x}, which is invariant under f 0 and transforms according to some irreducible representation f. It follows that
r
j,k
1,2,3,4 .
Taking Tk = T1 we see that j
1,2,3,4 ,
i.e. x = X(T 1)f, and consequently x E V. Since also x E Vi, it follows that x = 0, which contradicts dim span{x} = 1. We conclude that V = X; consequently G1 has precisely four different irreducible representations, which we denote by f·1 (i=1,2,3,4). Then (19) takes the form (16) . .Another way of writing (16) is in the form AAT = I, where A is the 4x4-matrix 1/2(a .. ). 1J This implies ATA = I, i.e. we also have (17). Denote by {x j I j = 1,2,3,4} the canonical basis of X : xj(Tk) = 0jk' Using this basis it is easily seen that 190
j
= 1,2,3,4 .
(23)
Expressing f a(T j ) with respect to the basis {f iii = 1,2 ,3,4} of X, and using (22), we find 4
trace fa ( T .) = I a.. . J i=l 1J
(24)
A.combination of (23) and (24) gives (18).
5.6.5. For each i 1
1,2,3,4 we define a synnnetric operator Pi E L(H) by
4
P. = -4 I a1·kTk · 1 k=l
(25)
It follows from (18) that
4
1 4 4 4 P. = -4 I I a·kTk = I 0lkTk 1 i=l k=l 1 k=l
I i=l
= Tl
=I .
(26)
We also have 1
4
1
4
TjPi = 4 k~l aikTjTk = 4 k~l aij(aijaik)(TjTk) 1
= a ij
4
4 ~~1 ai~T~ = aijPi
(27)
This implies 1
PkP. = -4 1
4
I j=l
1
~ .T.P. = -4
KJ J
1
4
I j=l
~J.al·J·Pl· = 0k1· Pl· .
K
(28)
We conclude from (26) and (28) that the Pi are orthogonal projections on mutually orthogonal subspaces Hi = Pi(H), which together span H : H = H1 ffiH 2 ffiH 3 ffiH 4 . If w E H is such that TjW = aijw for each j, then (25) and (16) show that Piw = w; this, together with (27) proves that
{wEH I TW=W, VwEG.} , 1
(29)
191
where the subgroup G.1 of G is defined by G. 1
=
{a. ·T·
lJ J
I j = 1,2,3,4}
.
So the elements of Hi are precisely those w E H which transform according to the representation r.; H.1 also consists of those wE H which remain invariant 1 under the subgroup Gi of G. 5.6.6. Remark. We will see in the remainder of this section that the introduction of the irreducible representations r.l facilitates very much the cal. culation of certain terms in the bifUrcation equation. This is mainly due to the simple form of the representations, the splitting of H into orthogonal subspaces associated with each r i , and also to the fact that the nonlinear term in the equation (1) is homogeneous. From a general point of view one may ask the question how useful group representation theory really is when dealing with nonlinear bifurcation problems with synnnetry. At first sight there seem to be a few severe handicaps: group representation theory is a linear theory, which can only be fully developed when considering complex representations. From the other side bifurcation theory is clearly nonlinear, and usually deals with problems in real spaces. The foregoing chapters seem to indicate that only concepts such as "equivariance with respect to some subgroup" or "invariant subspace" will be relevant. (See also Ruelle [185], Poenaru [174] , Golubitsky and Schaeffer " [ 75]) • However, once the problem has been pushed down to that of solving a finite-dimensional bifurcation equation, group theory can actually be of great help in two respects. First, it can sometimes be used to obtain a further reduction of the equations; in the next chapter we will give an example where the specific form of the representation is used to reduce a 5-dimensional problem to a 2-dimensional one. Second, the fact that the bifurcation equations remain equivariant may result in a serious restriction on the terms which can appear in the Taylor development of the bifurcation fUnction; here group representation theory is the appropriate tool when one has to find out which terms will vanish and which terms remain. This has been beautifully illustrated by the work of D.H. Sattinger ([195-197],[265] ). Also the remainder of this section illustrates this point. 192
5.6.7. From (26.) it follows that each p E H can be written in the form 4
I
p
i=l
p.
p.
1
1
=
P.p E H. • 1
(31)
1
We will use this form of p when dealing with the equation (1). Since P is an orthogonal projection on span ¢o c H 1 , it follows that (32)
Vp E H •
We put w
=
P
P¢o + v
E
lR ,
V E
ker P
(33)
and
in (1), and project on R(L), using I-P. We find Lv - AO~V + (I -P)C(p¢O+v) = v( (I -P)P1 +
4
I
i=2
(34)
p.) 1
For (p,~,v) in a neighbourhood of (0,0,0) the equation (34) has a unique solution v = v*(p,~,v) near v = O. 5.6.8. Lemma. There is a constant c > 0 such that for ciently small neighbourhood of the origin we have IIv*(p,~,v)1I
(p,~,v)
in a suffi-
3
.:;; c(lpl +Ivl) .
(35)
Proof. If no such constant c exists, then we can find a sequence {(vn ' Pn' ~n' vn ) I nEN}, converging to (0,0,0,0) as n -+ each n EN, (vn ,pn ,~n ,vn ) solves (24), IIvn I f 0 and
00,
and such that for
3
IPn l liVT -+ 0
asn-+
oo
•
n
Expressing that (vn,Pn'~n,vn) is a solution of (34), dividing by IIvnll and taking the limit for n -+ gives 00
193
vn
lim L IIvnll
n-+oo
0.
Since L has a bounded pseudo-inverse K v lim _n_ n-+oo II Vnll
=
R(L)
~
ker P, it follows that
°,
which is impossible.
0
5.6.9. The bifurcation equatio~. Using (32) and the solution v*(p,lJ,v) of (34) gives us the bifurcation equation for (1) in the form : (36)
From lemna 6 and the fact that C(w) is homogeneous of degree three it follows easily that F(O,O,O)
D F(O,O,O)
°
P
and
(37)
(38) (See lemna 2.4.25). We conclude that equation (1) satisfies the hypotheses (H1) and (H3) of this chapter. From (36) and (4.8)-(4.9) we see that DlJyoCO,O)
o
DvYO(O,O)
Dj) Y1 (0,0)
-1
DvY1(0,0)
(39)
The transversality condition (4.7) will be satisfied if (p,¢O) = (P1'¢O)/O. If this is the case, then, in a (j),v)-plane, the bifurcation set for (1) will have the generic cusp-form which we found in subsection 4.2. A necessary condition to have the generic situation is that Pl = P1p I O. This implies that when we consider normal loads having a particular symnetry (e.g. p E H2) then it may very well be that this symnetry prevents the generic condition to be satisfied. So, for certain symnetries of the normal load we expect to have non-generic bifurcation sets. In the remainder of this
194
section we will study how the bifurcation set depends on the components Pi (i = 1, Z, 3,4) of p; more in particular, we are interested in the non-generic case that p E HZ ffi H3 ffi H4 (i.e. P1 = 0), and in the transition between the ge·neric and the non-generic case (i.e. P1 small). In order to do so we will suppose that in equation (1) p E H has the form 4
P = EP1
+
2
p.
p.
i=Z
E
1
1
H. (i=1,Z,3,4) , E EJR. 1
We consider € as an additional parameter; the elements p.1 and we suppose that P1 is such that
E
(40)
H.1 are fixed,
5.6.10. Let us return to the auxiliary equation (34). If we use the form (40) for p (i.e. replace P1 in (34) by EP1)' put ~ = € = 0 and neglect the cubic term, then the remaining equation has the solution v* = \l o
4
2
i=Z
Kp. ,
(4Z)
1
where K is the pseudo-inverse of L. Therefore we put 4
v
= \l
I
i=Z
Kp.
+
~
~ E ker P
(43)
1
in (34); we obtain the following equation for ~ 4
I
L~ - AO]JA~ + (I-P)C(p¢O + \l
i=Z
Kp.
+ v)
1
4
= E\l(I-P)P1 + AO~\l
2
i=Z
(44)
AKp . . 1
We denote by v*(p,~,\l,€) the unique solution of (44); such solution exists for (p,~,\l) small and € bounded. A proof similar to the one of lemma 8 shows that we have the following estimate for ~*. 5.6.11. Lemma. There is a constant c > 0 such that for Ciently small neighbourhood of the origin we have
(p,~,\l,E)
in a suffi-
195
The bifurcation function
F(p,~,V,E)
F(p,~,V,E) = -~p+ (C(P¢o+v
4
I
takes the form Kp- +v*(P,~,V,E)),¢O) -Evk
i=2
1
°
5.6.12. Lemma. There is a constant c > such that all sufficiently small solutions (p,~,V,E) of the bifurcation equation F(p,~,V,E) = satisfy
°
(47)
Proof. If no such constant c can be found, then there is a sequence {(pn ,~n ,vn ,En ) I nEN}, converging to (0,0,0,0) as n -+ such that, for each n EN, we have F(pn ,~n ,vn ,En ) = 0, pn t and 00
°
I~ n I --2-+ 0 , IPnl
IEn II vn I 3 -+0, IPn l
°
Ivn I ---+0 Ipnl
asn-+
oo
•
3
Dividing F(pn'~n,Vn,En) = by IPn l , and taking the limit for n -+ us (C(¢O)'¢O) = 0, which contradicts (38). 0
00
gives
5.6.13. Let 4
FO(P,~,V,E) = -~p+ (C(p¢o+v
I
i=2
KPi)'¢O) -Evk .
( 48)
Since C(w) = iB(w,B(W,w)), where B(.,.) is a bounded bilinear operator on H, it is easily seen that there is a constant c > 0 such that ( 49)
Using (49) and the estimate (45) for v*, it is straightforward to show that IF(p,~,V,E)-FO(p,~,V,E)
I
~ c[(lpl+lvl)5 + (1~1+IEI)(lpl+lv\)31 , for some c > 196
° and for all
(p,~,V,E)
in a neighbourhood of the origin.
(50)
We now determine a more explicit form of the function (2.4.18) and (2.4.19) we find 332
FO(P,~,V,E) =aOp -~p-Evk+2pv +
4
I
i=2
FO(P,~,V,E).
Using
(B(KPi,B(¢O'¢O))'¢o)
4 4 pv 2 il2 jl2 [(B(KPi,B(KPj'¢O))'¢O) +
4
4
+~v3 I
~(B(¢O,B(KPi,KPj))'¢O)]
4
I
I
i=2 j=2 k=2
(B(Kp.,B(Kp.,KPk))'¢O)
(51)
J
l
As a consequence of symmetry considerations several terms under the summation signs in (51) will vanish. In order to study this in detail we will use the irreducible representations of the group G1 • 5.6.14. Let w. E H. n R(L); then TKw. = KTW. = Kw. for each T E G.• This shows l l l l l l that K maps H.l n R(L) into Hl.• Next, we have for each T E G : Vu,v,w E H •
TB(u,B(v,w)) = B(TU,B(TV,TW)) This implies that for w.
H., w.
E
l
l
J
E
H. and wk
J
E
(52)
Hk we have
(53) From this relation we see that B(w. ,B(w.,wk )) transforms under the operators l J T,Q, E G1 according to the representation f : G1 -+]{ given by ai,Q,aj,Q,ak,Q,
fi(T,Q,)fj(T,Q,)fk(T,Q,) ,Q, = 1,2,3,4 ;
(54)
this represenation is denoted by f· 19 f. 19 f k . The next table gives the prol J duct representations f. 19 f. as a function of f. and f. l
J
l
J
197
@
f1
f2
f3
f4
f1
f1
f2
f3
f4
f2
f2
f1
f4
f3
f3
f3
f4
f1
f2
f4
f4
f3
f2
f1
5.6.15. Let us now consider the different terms appearing in (51). First, since ¢O E H1, the table above shows that B(KPi,B(¢O'¢O)) E Hi for i=1,2,3. Since the subspaces Hi are mutually orthogonal, we have (B(KPi,B(¢O'¢O))'¢O) = O. Next, BCKPi,B(KPj'¢O)) transforms according to f i @f j , which equals f1 if and only if i = j. It follows that
the same conclusion holds for (B(¢O,B(KPi,KPj))'¢o). Finally, B(KPi,B(KPj,KI\:)) transforms according to fi@fj@f k ; f6r i,j,k E {2,3,4} this equals f1 if and only if (i,j,k) forms a permutation of (2,3,4). Taking all this together, we find the following expression for FO(p,~,v,E) (55) where b
and
198
4
I [IIB(Kp·'¢O)1I i=2 1
2
1
+-2(B(Kp.,Kp.),B(¢O'¢O))]' 1
1
(56)
c
=
(B(KPZ, KP3) ,B(KP4'
O, aElR} •
a are of class
(11)
C1 • 1vbreover
x(O,.\) = xO(.\,a(O,.\)) ,
(12)
f(a)xO(.\,a) = f,xO(.\,a) = xO(.\,a)
Va
271" f(lCP)x(P,'\) = f,X(p,,\) = -x(p,.\)
p
E
lR ,
= O,l, •.. ,k-l ,
(13)
(14 )
and a(-p,.\)
=
a(p,.\)
o
(15)
The theorem shows that for fixed ,\ near 0 the solutions of (1) near the origin are completely determined by the "bifurcation diagram" B.\, that is the subset of the (p,a)-plane given by B,\
=
((p,a) E U I either p = 0 or a = a(p,,\)}
(16)
This diagram is symmetric with respect to the a-axis. 6.4.4. Application: buckling of a circular plate. As an application of theorem 3 we consider the buckling problem for a clamped circular plate, subjected to a uniform radial pressure along the boundary, and to a small radially symmetric normal load. The equations for the equilibrium state of such a plate were discussed in section 2.4; we obtained the equation
(I-AA)w
+
C(w) = p
(17)
in the Hilbert space H = W~,2(Q), where Q = {(X,y)ElR2 1 x 2+il;
It is straightforward to verify the following properties of these operators: (i)
Pk is a continuous projection in H, for each k EN; (use an argument as in the proof of theorem 2.5.13); VkEN, VaE:ffi.
(27)
(iii) Pk is self-adjoint
if Hk = R(Pk ), then Pk is the orthogonal projection on Hk , and the subspaces Hk are mutually orthogonal
(v)
Vw E H
(vii) if E is a subspace of H on which r induces an irreducible representation of 0(2), then E C Hk for some kEN; if k = 0, then the representation is trivial; if k > 1, then E has a basis {u 1 ,u2} such that with respect to this basis r is given by (2). Since A commutes with the symmetry operators, it follows that VkEN,
(28)
that is, each of the subspaces Hk is invariant under the operator A. Let Ak be the restriction of A to Hk . Then Ak is a compact, self-adjoint and positive operator on Hk . Its characteristic values are precisely the Akj' j = 1,2, ... introduced in the previous subsection. 228
6.4.7. Now fix some characteristic value Akj , with k f 0. We want to solve equation (17) for (W,A,p) ina neighbourhood of (O,Akj,O) in Hx1RxHO' Let A = (1+a)Ak j; then (17) takes the form (1), with M(w,p,a) = ~jW - aAkjAw + C(w) - p .
(Z9)
(30) The group action on ker ~j is given by (Z). It is easy to verify the hypothesis of theorem 3; for example, if u E ker ~j \ {a}, then -AkjAu = -u ¢ R(~j)' which proves (H3). Theorem 3 gives us the existence of a radially symmetric solution WO(A,p) for each (A,p) near (Akj'O) in1R x HO; a11 other solutions appear in families of the form
for some (p,p) near the origin in1R x HO' The mappings w(p,p) and o(p,p) are smooth and have the properties described in theorem 3. 6.4.8. The bifurcation equation. In order to decide for'what values of (A,p) equation (17) has nonradially~symmetric solutions, we have to study the behaviour of the function o(p,p). Since o(p,p) is the solution of the equation (10)we first determine the function h(p,p,a). We assume that the radial function ukj(r) is normalized such that (u 1 ,u 1) = (u Z,uZ) = 1. It fo11ows from the transformation properties of u 1 and Uz that (u 1 ,u Z) = O. For the projections P and Q we take Vw E H •
(31)
The operator B equals -I in our example. Using (Z.17) and (8) we obtain:
L 1
h(p,p,a)
= a -
(u1 ,DWC(TpU 1+V*(TPU 1 ,p,a)). (u 1+Duv*(TPU 1,p,a)u 1))HdT (3Z)
ZZ9
Here v*(u,p,o) is the unique solution v
E
(ker ~.)1 of the equation J
Lk.v - OAk.Av + (I-Q)C(u+v) = p . J
(33)
J
It follows that v* (0,0,0) = 0, and, since C is a cubic operator v* (-u,-p,o) = -v* (u,p,o)
(34 )
Consequently h(p,p,o) will be even in p e HO' The solution a(p,p) of h(p,p,o) = satisfies
°
a(-p,p) = a(p,-p) = a(p,p)
V(p,p) .
The expression for h takes a somewhat simpler form for p v*
(0,
°,°) °we have
(35)
0. Since
=
1
V* (pu1'0,o) = p
fo
DuV* (TPU 1 ,0,0) .u 1dT
using the fact that C is cubic this gives
From this expression we obtain (36)
This implies (37) for all (p,p) in a sufficiently small neighbourhood of the origin. Let o(p) = a(O,p); then we obtain from (37) the following result. 6.4.9. Theorem. Let Ak . be a characteristic value of A, with k f 0. Then there is a neighbourho6d W of the origin in H, and a neighbourhood U of (Ak.'O) inlRxH O' such that for (A,p) E U the equation (17) has nonradially J
230
symmetric solutions w E W if and only if (38)
When this condition is satisfied, then (17) has exactly one family of nonradially symmetric solutions in W; this family has the form {r(a)W(A,p)
I aEIR}
for some W(A,p) E W. The function a(p) is smooth, with cr(O) = cr(p).
o and
cr(-p)
Proof. Given (o,p) near the origin inIRxHO' it follows from (37) that there is some p> 0 such that 0 = cr(p,p) if and only if 0> cr(p). This proves the theorem, if we define W(A,p) = w(p,p), where p> 0 is chosen such that A = (1 + cr(P,P))Ak.. 0 J
6.4.10. Remark. It is interesting to compare the foregoing results with the results of chapter 3 and more in particular with theorem 3.6.6. Indeed, this theorem implies that every small solution of (17) will remain invariant under r after application of an appropriate r(a); the relation (3) is nothing else T than the hypothesis (R) used in chapter 3. Considering solutions which are invariant under r , one obtains the reduced bifurcation equation (4). The T theory of this chapter shows that (4) consists of one single scalar equation, which can be handled if the "tr:ansversality condition" (H3) is satisfied. 6.4.11. A nonlinear boundary value problem. Let n be the unit ball in IR2, R> 0 and A the subspace of C3 (fi x[ -R,RD consisting of functions f(x,v) such that f(x,O) = 0 for all x E n, and f(x,v) = f(y,v) if IIxll = lIyll (we use the Euclidean norm in IR2). For each f E A we want to detennine 0 E IR such that the nonlinear boundary value problem ~v +
v
=
of(x,v) 0
=0
in n along
an
(39)
has nontrivial solutions. So this is a nonlinear eigenvalue problem. To bring 231
it in the form (1) we let X = {vE C2 ,a(i1) I vex) = 0, Vx E Clr2} and Z for some a E ] 0,1 [. The mapping M : X x A xli. -+ Z given by M(v,f,o)(x) = 6V(X)
+
Vx
of(x,v(x))
E
ri
CO ,a(ri) ,
(40)
is defined and of class C2 for all (f,o) E A xli. and for v in a neighbourhood of the origin in X. Our problem can then be written in the form (1). Moreover, Mis equivariant with respect to (0(2),r) if we use a representation r : 0(2) -+ L(Z) defined as in (18). Fix some fO E A and consider the linearized problem 6V +
v =
oDvfO(x,O)v
°
°
in r2 (41) along Clr2 •
For simplicity, let us also assume that Vx
E
i1
(42)
Since Clr2 is smooth, and DvfO(x,O) continuously differentiable, classical and generalized solutions of (41) coincide. Using the Hilbert space method of section 2.3, in combination with the projection operators Pk defined as in (26) (but with H = L2 (n)), one can show that for each kEN there is sequence {Ok_ I j = 1,2, ... } of positive characteristic values of (41); we have,. 0kj -+ J, as j -+ co. Avoiding accidental degeneracies, the corresponding eigenspaces will be one-dimensional if k = 0, and two-dimensional if k> 1. For k = 0, the eigenspace is spanned by a radially symmetric eigenfunction ~kO(r); if k f 0, then the eigenspace is spanned by functions u 1 E X and u 2 E X, of the form : u 2 (r,e)
= ~kj(r)sinke
( 43)
6.4.12. Let now °0 = 0kj for some k f 0; then ( 44)
also, since (41) is formally self-adjoint 232
(45) In (45) we use the bilinear fonn (u,v)
f~
=
Vu,v
u(x)v(x)dx
E
Z.
(46)
It is easily seen that the hypotheses (H1) and (HZ) are satisfied. For our example (H3) reduces to the condi hon (47)
which is satisfied because of (4Z). Application of theorem 3 then shows that for each f near fO in A the problem (39) has the following solutions near (0,°0) : (i)
for each ° near °0 there is a unique solution which is radially symmetric; since f(x,O) = 0 for each f E A, this unique radially symmetric solution is the trivial solution v = 0
(ii) for each sufficiently small P > 0 there is an eigenvalue Of(P) such that, for ° = 0f(P), (39) has a unique family of nonradially symmetric solutions, of the fonn
The function of(P) is of class solution vf(p) satisfies
C2
in P and f, and 0fO(O)
p = O,l, ... k-l .
( 48)
6.4.13. Now we modify the problem (39) adding a small inhomogeneous term 6V + of(x,v)
= g(x)
XE~,
(49) vex)
=
0
Here again we take ~
x
E
3~
.
{(xl ,x Z) EJRZ I xl Z+XZ Z< 1}, f
E A,
and we suppose that Z33
g is small and g
E
ZJL for some JL
E
N\ {O}, where (SO)
and (51)
For g = 0, (49) reduces to (39), and the equation is then equivariant with respect to the group 0(2); for g f one has only equivariance with respect to the subgroup ~JL' Fix some fO E A which satisfies (42); let 00 = 0kj' where 0kj is a characteristic value determined by (41), with k f 0. We want to find solutions (v,o) of (49) near (0,00)' for each (f,g) near (fO'O) inAxZJL' To find such solutions we have to solve the bifurcation equation
°
F(u,f,g,o) =
°,
(52)
where F: UxAxZJLxffi-+Uis of class C2 , U= span{u 1 ,u 2 }, F(O,f,O,o)
0,
DuF(O,fO'O,oO) = 0, DuDoF(O,fO'O,oO) = I U' Vs E 0(2) ,
F(r(s)u,f,O,o) = r(s)F(u,f,O,o)
(53)
and F(r(s)u,f,g,o) = r(s)F(u,f,g,o)
(54)
We will not give a complete discussion of the equation (52), but apply the ideas of theorem 2.12 and theorem 2.14 to obtain certain solutions which are a consequence of the equivariance (54). 6.4.14. In order to apply the approach of theorem 2.12 we need to have F(O,f,g,o) = for all (f,g,o). According to remark 2.22 this will be the case if we have
°
UEU
r(s)u = u
, Vs E
~JL
~
u =
°.
Using (2) we see that this will be satisfied if k / JL is not an entire nurnbeT; if we write k / JL = k1 / JL 1 , such that kl and JL 1 have no common divisors, this 234
means that £1 ~ 1. Now we look for 80 = r(aO)u 1 E S = {r(a)u 1 I aEIR} satisfying the condition (2.43) of theorem 2.14, where we should remember that the group .is ~£ (and not 0(2)). The condition will be satisfied if a O is s~ch that
i.e. when
(55) for some (j,m) E ~ x~. Indeed, when (55) is satisfied then {r(aO)u1,r(aO)u 2} forms a basis of U, and
Then (2.43) follows easily. Since r(-¥)u = u for u E U, and ~ = {m£1 + jk 1 I m,j E~}, (£1 and k1 have no common divisors) it follows that we can rewrite the condition (55) in the form aO E {Im+ij(mod tTI) I m,j E~} {2!:( k
m£ + jk 1£ 1) 1
{f·f1 I p
=
mod 2TI k I m,]'E~}
0,1, ..• , U 1-1}
Moreover, we have {2£TIj (mod tTI) I j E~} m£ +
= {2kTI(
jk
\ 1
1)modiflm,jE~} = {-¥.i Ip=O,1""'£1-1} 1
so the condition on a O becomes 2TI ..E..1 1 +2TI P I p--01 a O E {T',Q, p-O,1'···'£1-1 }U {TI f' yT'Y, ""'£1 -1} • 1 1 1 235
We conclude that the set of 80 E S satisfying the condition (2.43) contains 2£1 elements, and can be written in the form 2IT J2...)u I p-0,1,···"Q,1-1}u{r(T·r)f(rr)u _ 2IT P IT 1 I } {r(T·£ 1 1 p=0,1'···'£1- 1 1
1
1
= {f(S)u11 SElI£}U{f(S)f(iof)u11 SEL\} . 1
o 11 y, 1of £1 lS ° Plna
dd t hen K·I1 IT 1 = K IT mo d T·I1' 2IT 1 an d Slnce ° f(IT) K u = -u f or uE U, the set above takes for this case the form 0
Then the results of section 2 give us the following theorem. 6.4.15. Theorem. Under the conditions mentionned in the previous subsections, suppose that £1 t 1. Then the bifurcation equation (52) has, for each (f,g) E AXZ£ near (fO'O), at least the following nontrivial solution branches bifurcating from the trivial branch {(O,o) I IO-Okj I = 2'ff (w 1 (t),w2 (t))dt
(4)
Finally, if A E LORn) then we will denote by ker c A the kernel of the canonical extension of A to a linear operator over a;:ll. 6.5.3. Abstract fonnulation of the problem. Defire M
M(x,a) (t) = -:ic(t) + f(t,x(t) ,a)
Vt
E
XxlR.
~
lR •
Z by (5)
Then M is of class C2, M(O,a) = 0 for all a, and our pr~blem takes the form
M(x,a)
=
0 .
(6)
Let Dm = { = 0, VW* E ker(J* -~I)} - .
o
(21)
We call ~ E C an eigenvalue of J if ker(J-~I) is nontrivial; these,eigenvalues are also called the Fioquet expone~ of the 2n-periodic equation (12). The lemma shows that ~ E C is an eigenvalue of J if and only if 2 e 'f1l.l is an eigenvalue of the monodromy matrix C; these corresponding eigenvalues e2n~ are called the Fioquet m~p~~ of (12). This also shows that if ~ is an eigenvalue of J, then so are ~+ii and y+ii, for any i E~. 6.5.7. Now we will make some further hypotheses on the equation (1). In order to do so we define, for each cr E R, the operator J(cr) E L(XO,ZO) by c
(J(cr)w) (t) = -wet) + A(t,cr)w(t)
Vt
E
lR •
c
(22)
We have J = J(O), and J(cr) has similar properties as J. Let J (0) = DoJ(O); Jo(O) is an element of L(X~,Z~) which can in fact be extended°to all of z~, 240
and which is explicitly given by Vt
E
lR .
(23)
We make the following assumption (H)
(i) There is some k EN such that i~ is a Jcr(O)-simple eigenvalue of J (see definition 6.3.1). By the remark after lemma 6 we may suppose 0";; k..;;
m 2.
(ii) J has no eigenvalues of the form i~, with k' EN, 0 ..;; k' ..;;~, k' =f k. The hypothesis (H)(ii) is called a nO~e60nanee eondition; using (19) it says that the only eigenvalues A of C such that Am = 1 are given by
exp(±2~ik). Let us explain now in some more detail the meaning of (H)(i). Let 110 = ~i. Then (H) (i) implies first of all that dim ker(J-110I) = 1, which, by (19), translates into dim kerc(C - e 21T11o I) = 1. Let 1:;0 E cen and 1:;0 E cen be such that (24)
Define 1:;,1:;* E x~ by (25)
then we have ker(J-110I) = span{l:;}
,
ker(J*-00I) = span{I:;*}
(26)
and R(J-110I)
=
{wE z~
I = O}
•
(27)
The condition that 110 is a Jcr(O)-simple eigenvalue of J means that Jcr(O) I:; ¢ R(J-llOI), i.e. f 0 , cr
(28) 241
or more explicitly 1 -2 If
21f
J
0
(c:;* (t) ,D A(t,O) c:;(t))dt f 0
(29)
a
We will normalize C:;o and C:;O by the condition a
=
2 •
(30)
6.5.S. Remark. In other treatments of the problem of bifurcation of subharmonic solutions (see e.g. Iooss and Joseph [255]) one usually assumes that e21f~o is a simple eigenvalue of C, which means that we have (24) and C:;o ¢ Rc(c-e21f~OI), i.e. (c:;0'C:;0) f O. Then one normalizes by the condition that (sO'C:;O) = 1. It is easily seen that this implies ~ 1. Let us show that this implies (H) (i) under an additional condition. Let C(a) be the monodromy matrix for the linear 21f-periodic equation :ic
=
A(t,a)x .
(31)
lffider the condition (c:;0'C:;0) f 0 one can show (for example by using a complex version of the Crandall-Rabinowitz theorem given in section 3) that C(a) has a continuously differentiable eigenvalue branch e2~(a), such that ~(O) = ~O. Then ~(a) is an eigenvalue of J(a), and one can construct c:;(a) E xO such c that c:;(0) = c:; and J(a)c:;(a) = ~(a)J(a) Differentiating (32) at a obtain a
=
Va •
(32)
0, and taking the inner product with c:;*, we
D ~(O) , a
(33)
since (so,so) f 0 implies (H) (i). 6.5.9. We now return to our description of ker L, using (17). It follows
from our hypothesis (H) that 2ni 9, u ker (C - e m I)) O.;;;t<m c _ 2nik 2nik span(kerc(C - e m I) u kerc(C-e m I))
span (
Define X E Xc and X* E Xc by .k 1-t x(t) = ~(t)sO = e m set)
x* (t)
(34)
Then we see from (17) that ker L = {Re(zx)
I ZE(C}
,
ker L* = {Re(ZX*)
I zE(C}
•
(35)
Let kim = k/m1 , where kl and m1 have no common divisors. (Take m1 1 if k = 0). Since set) is 2n-periodic (i.e. rs = s), it follows from (34) and (35) that Vu E ker L ,
(36)
that is, the elements of ker L are all 2'ITID1-periodic. Then we know from the theory of chapter 3 that all sufficiently small solutions of (6) will also satisfy rmlx = x, i.e. all such solutions are 2'ITID1-periodic. So we can replace k and m in the foregoing theory by kl and m1• This implies that we may assume that k and m have no common divisors, and that 0 .;;; k < We will consider now several cases, depending on the value of m.
-z.
6.S.10. The cases m = 1 and m = 2. If m = 1, then k = 0 and the hypothesis
(H) says that C has an eigenvalue equal to 1, and that the corresponding eigenspace is one-dimensional. We may take SO and SO to be real vectors, 243
and it follows that dim ker L = 1. We have a bifurcation problem from a simple eigenvalue. Because of the condition (28) it is possible to apply the Crandall-Rabinowitz theorem discussed in section 3, to obtain one single branch of 2TI-periodic solutions of (1) bifurcating from the zero solution. From now on we will suppose k f O. If m = 2, then k = 1, and C has an eigenvalue equal to -1, and the corresponding eigenspace is again one-dimensional. The elements of ker L satisfy u(t+2TI) = -u(t). As for the case m = lone has here also a problem of bifurcation from a simple eigenvalue, on which one can apply theorem 3.3. One finds a single branch of 4TI-periodic solutions bifurcating from the trivial solutions. This branch has the symmetry given by the second part of theorem 3.3. We leave the details to the reader. m 21Tk. 6.5.11. The case m ~ 3. When m ~ 3 and 0 < k < 2' then exp(-m-l) is complex, and it follows from (35) that dim ker L = 2. We can define a basis {u 1,u 2} for ker L by 1
-
1
u 1 = Re X = 2(X+X) It also follows from (34) that 2TIk. -l fX = e
m
X
-
1m X = 2i(X-X) .
fX'
(38)
21Tk.
-l
e m X.
(39)
Using (38) we see that
(40)
We conclude that the representation f of Dm is irreducible on ker L; moreover, (40) shows that f is orthogonal with respect to the basis {u 1 ,u 2} of ker L. More generally, we have = The range of L is given by 244
Vu,v E ZC
(41 )
R(L) =
{WE
Z I <X' ,w> =o} .
(42)
Using (5),(9),(23),(30) and (34) it is easily seen that <x* ,D D MeO,O)x> x
°
2,
( 43)
while
4TIk.
1
(-
m- 1 - - ] \' m L
m j=O
e
)<X*,D D M(O,O)x> °x
°.
It follows in particular that <X* ,D D M(0,0)u 1> °x R(L).
( 44) 1, i.e. D D M(0,0)u 1 x
°
~
6.5.12. Projection operators. We define projections P and Q, respectively on ker L and on a complement of R(L), as follows Pw = Re«X* ,D D M(O,O)w>X)
°x
and
Vw E X
( 45)
Vw
(46)
E
Z •
Using (43) and (44) it is easily verified that P and Q are indeed projections on the appropriate subspaces, while also pr
Qr = rQ .
rp
If we let w.
(47)
D D M(O,O)u.1 (i= 1,2), then also
lOX
Bu.l O = QDX D M(O,O)u.1 = Qw. = w. 11
(i = 1,2) .
( 48)
Using the foregoing definition in the general fonnula (2.17) for the bifurcation function F, we obtain for our problem F(u,o) = Re( <X* ,M(u+v' (u,o) ,o»X)
V(u,O) E
ker L xJR ,
(49) 245
where v'(u,a) is the unique solution of the appropriate auxiliary equation. From the form (49) for F, in which also u = Re(zX) for some z E C (see (35)) it seems reasonable to use complex coordinates when discussing the bifurcation equation, which is an equation in a two-dimensional space. We introduce now such coordinates. 6.5.13. Complex coordinates. Consider the following mapping from ker L = U into C, the complex plane (SO)
ljJ is a bijection with inverse Vz = x+iy
E (C •
(51)
So ljJ(Re(ZX)} = z, Vz E C. The mapping ljJ is a linear isomorphism if we consider both U aYld (C as two-dimensional ILeal vectorspaces. To each mapping F : U -r U we can associate a mapping G : (C -r (C defined by G = ljJoFoljJ-l; conversely, to each G ; C -r(C there corresponds a mapping F : U -r U given by F = ljJ-1 oGoljJ . When doing calculus on the mapping G (for exampIe, finding the Taylor expansion of G) one has to keep in mind that (C is considered to be a real vectorspace. For example, one has Vb E C •
More practically, one first defines F : U -r C such that Feu) Re CF(u}x}, Vu E U; this F is given by F = ljJoF. Then G(z) = F(Re(zx)), Vz E(C, and if F (and consequently F and G) are sufficiently smooth, then we have, for each h = h 1+ih Z E (C DG(z)h = DF(Re(zx))·Re(hx) = DF(Re(zx)).(h 1u 1-h zu z) 1
~
~
= ~ DF(Re(zx))u 1 + iDF(Re(zx))uZ]Ch1+ihz) 1
~
~
+ Z[DF(Re(zX))u 1 -iDF(Re(zx))uZ ](h 1-ih Z)
= DzG(z)h + D-G(z)h , z 246
(52)
where the functions DzG : a:::
-+
a::: and DzG : a:::
-+
a::: are defined by (53)
and 1
~
~
DzG(z) = "L[DF(Re(zx))u 1 - iDF(Re(zX))u 2 1 • These functions are associated to the mappings given by
(54)
Dl : U -+
U and DzF
U -+ U
DzF(u) = Re(DzF(u)x) ,
(55)
where (56) and (57)
The form (52) of DG(z), in which DzG(z) and from a::: into itself allow us to consider higher tiating the functions DzG and DzG in precisely This gives us for example the following Taylor class 2< : G(z+h) = G(z) +
k
I
1
DzG(z) are again functions derivatives of G by differenthe same way we did with G. formula if F (and G) is of
.
""""'T(hD + hD- )JG(z) + R(z,h) , j=l J . z z
where R(z,h) = o(lhl k ) as Ihl
-+
(58)
o.
6.5.14. The bifurcation function. From (49) we see that in the complex coordinates just introduced our bifurcation function will take the form G(z,cr) = <x* ,M(Re(zx) + v* (Re(zx) ,cr) ,cr»
, V(Z,cr)
E
a::: xJR •
(59)
We know from the general theory that G(O,cr) = 0 for all cr, while DG(O,O) = 0 and DcrDG(O,O) = IC. (By D we denote differentiation in the variable z, as described in subsection 13). Moreover, the function G is equivariant, i.e. 247
= rG(z,O'), where r : (C -+ (C is the action on (C corresponding is easily seen from (39) that this action is given by
we have G(rz,O') to (40). It
2"k.
-1
rz
e m z
VzE
(60)
(C •
k and m h ave no cornmon d1V1sors, we h ave fIe. · S1nce Lm] ( mOG'·1) Ii J- c~ [ki+rnQ, , ' . } rj, lmod 1) i J, 9, E:2. == mI j == 0,1, ... ,m-l}. Consequently we have
l-rn-
'L
2Tri
2"i
Gee m Z,G)
V(z,O')
== C m G(z,o)
(61 )
This equivariance condition strongly restricts the possibilities for the form G can take, as we will show now. Suppose t,.fJat M in (1) is of class CP ; then also F and· G will be of class Cp . Expanding Gez ,G) around the point z '" 0 we obtain from ~
G(z,G)
\'
==
-,Q,
a· (o)z)z
L.
l~j+x,~p
J 9,
+ R",< (z,O') ,
(62)
u
where R (z,o) == o(lzI P ) as I zl -+ O. The f1.L'1ctions a- n :]R -+ (C are, up to a p . ] Iv constant, given by DJD9~G(0 ,G). It follows from the equivariance condition
z z
(61) that a·nCG) call be different from zero only if J.>V
j
~ +
1 (mod m) .
(63)
We now consider separately the cases m 6.5. 15. ]ne case m
==
3. When m
==
==
3, m
4 and m > S.
3 (and consequently k
==
1) then the rule
(63) gives us the following fonn for the bifurcation function
G(z,G)
==
-i: a(o)z + S(o)z + R3 (z,0) ,
(64)
where R3 == 0(lzI 3 ) if we assume fin (1) to be of class C3 . Moreover (t(O') 0' + O( 10'1 2 ), and a straightforward calculation shows that
S(O)
==
1
g<X*
_ 1 1 -8 "b-n
248
7_.
- -
,D~M(O,O)(x,x»
r6 "
2
Jo Cx' (t) ,D/(t,O,O) (xCt) ,xCt)))dt
(65)
We will assume that S (0) f O.
(z,a) of G(z,a) o in a sufficiently small neighbourhood of the origin we have 101 < elzl As in subsection 2.19 one proves that for each solution
for some constant e. This allows us to put (66)
P].1
a
in (64); since (p,¢,]J) and (-p,¢+TI,-].1) correspond to the same point (z,a), we will assume ].1 .;;;' O. Bringing (66) into (64), we can divide by p2 in the resulting equation, and obtain H(p,¢,]J)
=~
G(pe i ¢,p].1) = ].1e i ¢ + S(0)z-2i¢ + R(p,¢,].1) = 0 ,
(67)
P
where R(p,¢,]J) is of class e 1 and such that R(O,¢,].1)
= O.
In order to solve
(67) for small p, we have to find a solution of the reduced bifurcation equation H(O,¢,].1)
= 0,
which takes the form (68)
When SCO) f 0 then this equation has the solutions (¢O + 23TIj , ].10)' with ].10 IS(O) I, j
E 7l
=
and an appropriate ¢O. l'vloreover : D(¢,]J)H(O,¢O + 23TIj ,].10) is an
isomorphism from]R2 onto
ce.
We can apply the implicit function theorem on
(67) to obtain the following.branches of solutions: (¢(p) + 23TIj,].1(p)), where ¢(p) and ].1(p) are of class e l , ¢(O) = ¢O' ]J(O) = ]JO and j E 7l. We leave it to the reader to construct from these solutions the corresponding branches of 6TI-periodic solutions of (1). Remark that the solutions corresponding to different values of j (only j
=
0,1 and 2 give different solutions) can be
obtained one from the other by a phase shift over 2TI or 4TI, i.e. by application of the symmetry operator
r.
We conclude that W1der the hypothesis CH) with m
= 3 (and k =
1) there
will generically (i.e. when SCO) f 0) be bifurcation of 6TI-periodic solutions. 6.5.16. The case m = 4. When m = 4 (and k
1) we find from (63) that the
bifurcation function takes the form G(z,a)
2-
a(a)z + y(a)z z +
-';
oCa)z~
+ R4 (z,a) ,
(69) 249
where R4 = 0(lzI 4), assuming that f in (1) is of class C4 . We still have a(o) = a + 0(101 2), and one can express yeO) and 0(0) in the form of integrals, involving X(t), X* (t), derivatives of f(t,x,o), and the first approximation of v*, which can be obtained by solving an inhomogeneous variant of the linear equation (12). We will not give the details here, but only discuss what kind of solutions we can expect. All small solutions of G(z,o) = 0 will satisfy 101 ~ Clzl2. Therefore, we put a = llP
2
(70)
in (69). Dividing by p3 we obtain the equation H(p,¢,ll)
= lle i ¢
+ y(O)e i ¢ + 0(0)e- 3i ¢ + R(p,¢,ll)
where R is of class C1 and R(O,¢,ll) has the form H(O,¢,ll)
=
(71 )
O. The reduced bifurcation equation
Cll+y(O))e i ¢ + 0(0)e- 3i ¢
=
0
(72)
1his equationwill have solutions (ll,¢) EJR2 if and only if 10(0)1 ;;;.IIm y(O)I. If 10(0)1> 11m y(O)1 then there are two sets of solutions : (¢~1) +-¥j,1l6 1)) and (¢~2) +-¥j ,1l6 2)); at these solutions the Jacobian for (72) is different from zero, and we can apply the implicit function theorem to obtain ~o times four branches of 8n-periodic solutions. The four branches correspond to different values of j, and correspond to phase shifts over multiples of 2n. Mbreover, the bifurcation function G(z,o) is odd in z (by lemma 2.16), so that ll(P) is even in p; this gives some supplementary symmetry in the bifurcation diagram. We conclude that for m = 4 there will only be bifurcation of 8n-periodic solutions if the condition 10(0) I;;;. 11m y(O)1 is satisfied. 6.5.17. The case m;;;' 5. When m;;;' 5 then the bifurcation function has the form G(z,cr) = a(o)z + y(o)z 2-z + R4 (z,0) ,
250
(73)
with R4 = O(lzI4). Again we have 101 ~ clzl 2 for small solutions of the bifurcation equation, so that we can use the rescaling (70). This leads to the following reduced bifurcation equation (74) Generically yeO) will not be real, in which case (74) has no solution (~,¢) E IR2. For m ;;;, 5 there can only be bifurcation of subharmonic solutions if yeO) EIR, and even then some further conditions have to be satisfied. One speaks about weaQ h~onan~e when there is indeed bifurcation. 6.5.18. Comments. It is an interesting exercise to find all possible irreducible representations of the group Dm ; such representations are 1-dimensional if m ~ 2, and 1 or 2-dimensional if m;;;' 3. For fixed m one finds precisely those representations which we found in the foregoing theory by restricting r to ker L, and by giving to k all possible values. The hypothesis (H) corresponds precisely to the irreducibility and the transversality condition in the general theory of section 2. Moreover, when the representation is 2-dimensional (i.e. when m;;;' 3 and k appropriate), then the equivariance of the bifurcation function gives us the selection rule (63), which is sufficient to discuss the qualitative bifurcation picture. One can find more details on bifurcation of subharmonic solutions in the books of Iooss [254] and Iooss and Joseph [255]. 6.6. A BIFURCATION PROBLEM WIlli O(3)-STI1t-.1ETRY 6.6.1. The problem. Let Q = {xEIR312 x 1+x 22+x32 < 1} be the unit ball in IR3, and consider the following boundary value problem L.v + ~f(x,v,A)
o
X E Q
(1)
vex)
=
0
x E
i'i xIR x fl
We assume that f f(x,O,A)
o
+
3Q
IR is of class C3 ,
V(X,A)
(2)
251
and f(x,v,A)
=
f(y,v,A)
if
II xII = IIyll ,
(3)
i.e. f is rotationally symmetric with respect to the x-variable. A is any Banach space, and ~ is considered as a nonlinear eigenvalue. We want to determine, for each A near 0, corresponding values of ~ for which (1) has a nontrivial solution. To bring the problem in the form (2.1) we let X = C~ ,Ci.(il) and Z = CO ,Ci.(il) for some Ci. E 10,1 [, and we define M : Xx A x]R -7- Z by M(V,A,~)(X)
= ~v(x) +
~f(x,v(x),A)
Vx
E
11 ,
(4)
where vEX. Then our problem has the form
o.
(5)
We can define a representation r -1
(r(R)v)(x) = vCR x)
0(31
-7-
L(Z) of 0(3) over Z by 0(3)
VR E
(6)
Then it is easily seen that the equation (5) is equivariant with respect to
(0(3) ,r). 6.6.2. The linearized problem. Let L(~) = DvM(O,O,~). In order to determine ker L(~) we have to solve the linear boundary value problem ~u+~Dvf(x,O,O)u = 0
X E Q
u(x) = 0
X E
(7) 3Q
One knows from the general theory of such equations (see e.g. [711 and section 2.3) that (7) has nontrivial solutions if and only if ~ belongs to a countable set of eigenvalues for the problem (7). If ~O is such an eigenvalue, then ker L(~O) will be finite-dimensional. Moreover, since the problem (7) is equivariant, the restriction of r to ker L(~O) will give a finite-dimensional representation of 0(3). Generically, one may assume that 252
this representation is irreducible. Then it follows from the theory of section 2.6 that dim ker L(vO) = 2£+1 for some £ EN, and that ,the elements of ker L(vO) transform under the symmetry operators r(R) in the same way as the spherical harmonics of order £. Using the completeness of the spherical har? monics on L2(S~) one can show that the elements of ker -L(VO) , when written in spherical coordinates, must have the form u(r,e) = ~O(r)y£(e), (r ~ 0, e E S2), with a fixed radial symmetric function ~O(r) and Y£ E U£, the space.of spherical harmonics of order £ (see (2.6.42)). Because of_ the foregoing one can _label the eigenvalues as ]J£,J (£ EN, j EN) . . with dim ker L(]J£j) = 2£+1, and ker L(]J£j) = {~£j(r)Y£(e) radial function ~£j(r) depending on £ and j.
I Y£EU£},
for some
6.6.3. Let us fix some eigenvalue ]J£" and study (1) for]J near ]J£,. To do _ J J so we put]J ]J£j+a and M(v,A,a) = M(v,A,]J£j+a). We have to solve the equation M(v,A,a)
a
(8)
for (v,A,a) near (0,0,0). We have L = D~(O,O,O) = L(]J£j); we know from the general theory for equation (7) that L is a Fredholm operator with zero index, and by the preceding considerations we may assume that the restriction of r to ker L is irreducible. Therefore the hypotheses (Hl) and (H2) of section 2 are satisfied •. As for (H3), this will be satisfied if (9)
for some u E U = ker L. However, it follows from (7), with ]J = ]J£j' that for U
E ker L(]J£j) we have
since ]J = 0 is not an eigenvalue. It follows that (9) is satisfied. The theory of section 2 shows that the problem (8) reduces to the bifurcation equation Feu,A,a) = 0
(10)
253
where F : Ux AxlR -+ U is defined and of class C2 in a neighbourhood of the origin, and has the following properties : F(O,A,a) = 0
V(A,a) ,
(11)
DuF(O,O,O) = 0
(12)
and F(r(R)u,A,a) = r(R)F(u,A,a)
VR
E
0(3) .
(B)
One could write down a precise expression for F; however, the properties (11)-(13) will be sufficient for the qualitative discussion which follows. 6.6.4. Bifurcation of axisymmetric solutions. Let us first look for axisymmetric solutions of (1); these are solutions (vl,A,~) such that, for a certain vector a E lR3 \ {O}, we have r(R)v 1 = v 1
VR
E
G a
(14)
where Ga = {RE 0(3) I Ra = a} is the isotropy group of the vector a. One can find some Rl E 0(3) such that R1a=ve 3 ; then v = R1v 1 will also be a solution, and satisfy Va
E
lR .
(15)
Conversely, let (V,A,~) be a solution of (1) satisfying (15).Then, for any Rl E 0(3), v 1 = r(R1)v will also be an axisymmetric solution, with axis defined by a = R1e 1 • This shows that it is sufficient to find solutions (V,A,~) satisfying (15). By the Liapunov-Schmidt reduction we have to find solutions (U,A,~) of the bifUrcation equation (10) which satisfy r(~3(a))u = u for all a E lR. It follows from corollary 2.6.29 and the irreducibility of rover U that dim{uEU I r(~3(a))u=u, VaElR} = 1. If this subspace is spanned by u 3 E U, then the condition (2.39) of theorem 2.12 is satisfied for 80 = u 3 • That means, we have F(pu3 ,A,a) = ph(p,A,a)u 3 , with h(O,D,O) = 0 and Dah(O,D,O) =1. The equation h(p,A,a) = 0 has a solution a = a*(p,A), by the implicit function theorem. This gives the following result. 254
6.6.5. Theorem. For each eigenvalue 11Q,j of (7) there exists a map 0* : :JRxA +:JR, defined and of class C1 in a neighbourhood of the origin, such that for each sufficiently small (p,A), with p> 0, and for 11 = 11Q,j +0*(p,A), problem (1) has a family of nontrivial axisymmetric solutions {r(R)v(p,A)
I REO(3)}
;
one has V(p,A) (¢3(a)x) = V(p,A)(X), Va E:JR, and V(p,A) = O(p). In case Q, = or Q, = 1 these are the only nontrivial solutions of (1) near (O,O,l1Q,j). In case Q, = 0, all these solutions are rotationally symmetric. r roo f. It remains to prove the last part of the theorem. For Q, = or Q, = 1 all u E U are axisymmetric, as can easily be seen from the definition of spherical harmonics. In case Q, = we have dim U = 1 and all u E U are rotationally symmetric. Then the result follows from the theory of chapter
°
°
°
3.
0
If we want to find non-axisymmetric solutions, we have to consider eigenvalues 11Q,j with Q, ~ 2. For such cases we have dim U = dim UQ, ~ 5. In the remainder of this section we will discuss the case Q, = 2. 6.6.6. The case Q, = 2. When Q, = 2, then dim U = 5, and (10) forms a fivedimensional problem. However, we can use the orbit structure of U under the action of 0(3) to reduce this to a two-dimensional problem. Our presentation partly follows Golubi tsky and Schaeffer [ 251 ]. Let us first look at the structure of the second order spherical harmonics. Each quadratic pol)'llomial H2 : :JR3-+ :JR can be written in the form H2 (x) = (x,Ax), where (x,y) = i~1 xiYi and A E LOR3 ) is symmetric: AT = A. The condition LH 2 (x) = 0, appearing in the definition of a spherical harmonic (see section 2.6), then reduces to trace A = 0. When we define
v = {AE LOR3 ) I AT =A and tr A=O} ,
(16)
the foregoing implies that there is an isomorphism between U and V. the group 0(3) is represented by r(R)A = RART
VR E
0(3)
VA
E
V .
On
V
(17)
255
Using the isomorphism between U and V, we can consider F as a mapping from V x AxJR into V, with properties corresponding to (11) - (13) . Let {e 1 ,e 2 ,e 3} be the canonical basis ofJR3 , and let D be the subspace of all diagonal A E V, i.e. D
= {A E V I <e. ,k . > = <e ~ ,Ae. >8 .. , i, j l
J
.~
l
lJ
=
1 ,2,3} .
( 18)
Since every synnnetric A E L(m3) can be diagonalized by an appropriate orthogonal similarity transformation, it follows that V
=
{r(R)A I AE D, RE 0(3)}
(19)
This implies that each orbit of solutions of the bifurcation equation has a nonempty intersection with D; consequently, it is sufficient to find solutions of (10) belonging to the subspace D. Let La be the 8-element subgroup of 0(3) containing those R E 0(3) of the form Re.l = E..e. (i = 1,2,3), with E..l = ±1. Then we have l l D
{AEVI r(R)A=A, VRELa} and
It follows that for A E D and R E r(R)F(A,A,O)
La
La
= {RED(3)
I r(R)A=A,
¥AED} (20)
we have
F(r(R)A,A,O) = F(A,A,O)
we conclude that F maps Dx AxJR into D. In the two-dimensional space D we will use coordinates (x,y) E JR2 defined by the isomorphism \~ : JR2 -+ D given by (Rez tjJ(x, y)
ljJ(z)
0
0
0 .21T lRe(e 3 z) 0
0
a
Re(e
.21T -l3
(21) z)
where, for notational convenience, we identify z = (x,y) EJR2 with z E~. Using this isomorphism (10) takes the form
256
x+iy
H(Z,A,cr) =
a,
(ZZ)
where H :]RZ x1\.x]R -+-]RZ is defined by H(Z,A,cr) = 1)J-1 p (1)J(Z),A,cr). The mapping H has the following properties, corresponding to (11)-(1Z) (Z3) . As for the SyrIJretry properties of H, let L = {RE 0(3)
I r(R) (D)
subgroup of syrrnnetry operators leaving the subspace D invariant; normal subgroup of
= D} be the
La is a
L. The quotient group LILa has 6 elelOOnts, which can be
identified with those RE 0(3) which merely interchange the coordinate axes, i.e. which have the form Rei = ev(i) (i= 1,Z,3) for SOIOO peIiIlUtation v of {1,Z,3}. The action of such symmetry operator on SOIOO A E D consists in a corresponding peIiIlUtation of the diagonal elelOOnts of A. Via the isomorphism 1/1 the corresponding action on lRZ ~ CC is generated by the mappings Q : CC -+- CC and T :
(C
-+- C given by .Z7f
QZ =
e13z
TZ
Vz
Z
(Z4)
E C •
Let ~3 = {I,Q,QZ,T,ToQ,ToQZ}; (for the notation ~3' compare (Z4) with (4.51)). Then the equivariance of H can be expressed by H(YZ,A,cr) = yH(Z,A,cr)
Vy
E ~3
(Z5)
•
Z Z k+4 . 6.6.7. Lemma. Let H : lR x1\. xlR -+-]R be of class C ,and suppose H satlsfies (Z5). Then there exist Ck-functions p : lRZ x 1\. x]R -+- lR and q : lRZ x1\. xlR +
lR such that (i)
H(Z,A,cr) = p(Z,A,cr)Z + q(z,A,a)z-Z
(ii) p(YZ,A,cr) = p(Z,A,cr)
Z
q(YZ,A,cr) = q(Z,A,cr) ,
Proof. Define h : lR x 1\. xlR -+- lR by
(Z6)
Vy
E ~3
•
(Z7)
.
h(z,A,cr) = -yH1(Z,A,cr) + xHZ(Z,A,cr) ,
(Z8)
where H = (H 1 ,H Z). It follows from (Z5) that Z57
h(cSz,A,o)
=
h(z,A,o)
, h(n,A,o)
=
-h(z,A,o) .
From the second of these relations it follows that h(z,A,o) i.e. if z = So we have
z.
1
(29)
o if z
(x,O),
-
h(z,A,o) = yh 1 (z,A,0) = zr(z-z)h 1(z,A,0) Now 1 h(z,A,o) = h(cSz,A,o) = 2i(cSz-cSz)h 1 (cSz,A,0) ;
z,
since, for z = we have cSz-cSz = (cS-8)z f 0 (except when z that h 1(oz,A,0) = 0 if z = as before, we conclude that
0), it follows
z;
1
-
h 1(cSz,A,0) = zr(z-z)h 2 (z,A,a) . Another application of the same argument gives us finally i 3-3 h(z,A,a) = Z(z -z )q(z,A,o) ,
(30)
for some q : JR3 x II xJR -+ JR satisfying the condition (27). ~ -2 Let now H(z,A,o) = H(z,A,o) - q(z,A,o)z ; then it follows from (28) and (30) that yH1 = xH 2, which implies that H1 (z,A,a) = p(z,A,o)x and H2 (z,A,a) = p(z,A,o)y for some p : JR2 x II xJR -+ JR, i.e. H(z,A,o) = p(z,A,o)z. This gives us (26), and also the first part of (27) follows immediately. 0 6.6.8. Now we bring the form (26) in the equation (22), put z = pe i8 , multiply by split into real and imaginary parts, and devide by p2, respectively p3; this gives us the following equations for nonzero solutions :
z,
p(pe i8 ,1..,0) + pq(pe i8 ,1..,0) cos38 i8 q(pe ,1..,0) sin38 =
0,
°.
(31)
(32)
°
Now it follows from (23) that p(O,O,O) = and Dop(O,O,O) = 1. We can apply the implicit function theorem to solve (31) for 0; we obtain 0 = a*(p,8,A), with 0*(0,8,0) = and 0*(P,±8+ 2;k,A) = o*(p,e,A), k = 0,1,2.
°
258
The equation (32) is satisfied for e = 2;k, k = 0,1,2. This corresponds to the lines y = 0 and y = ±l3x inR2. It is easly seen that these solutions of (31)-(32) generate precisely the branch of axisymmetric solutions described in theorem 5. 6.6.9. Non-axisymmetric solutions. For non-axisymmetric solutions we have sin3e f 0; for such solutions (31)-(32) reduce to
p(z;A,O') = 0
q(z;A,O')
=
0 .
(33)
Again the first of these equations can be solved' for a = a(z,A), with a(O,O) = 0 and a(yz,A) = a(z,A), Vy E ~3' by (27). It is also clear that for solutions of (33) we have a(pcose,psine,A) = O'*(p,e,A). Bringing the solution a(z,A) of p = 0 into the second equation q = 0, we obtain a final (scalar) bifurcation equation r(z,A) = 0 ,
(34)
with r : R 2 x/\ -+R given by r(z,A) = q(z,A,a(z,A)). The smoothness of r depends on the smoothness of the original problem, and, moreover, r is invariant under the symmetry operators r(yz,A)
=
r(z,A)
(35)
Using polar coordinates, it is easily seen that r(z,A) has the form (36)
r (z, A)
for appropriate functionals a,S: /\ -+R. If a(O) f 0, then (34) has no solutions (z,A) near the origin, and in this case there is only bifurcation of axisymmetric solutions, as given by theorem 5. Now suppose that a(O) = 0 and S(O) f 0; we will assume S(O) > 0; the case S(O) < 0 is similar. Consider the equation T(z,A,a) _ a+S(0)p2+[r(z,A)-a(A)-s(0)lJ
0
(37) 259
it coincides with (34) for a = a(A). There isa neighbourhood of the orIgIn in 1{2 x Ax1R and a constant C > such that solutions of (37) belonging to that neighbourhood satisfy p2 < ci al. Therefore we use in (37) the rescaling
°
n>o.
P
(38)
Dividing by ~2 in the resulting equation, we find
±1
+
S(0)n 2
+
O(~)
°.
(39)
If a > 0, then one has the plus sign in (39), and there are no solutions n when ~ = 0; consequently, there are also no solutions for ~ near 0. If a < 0, then one has the minus sign, and one can solve for. n; one finds n = n*(e,A,~), with n* (e,O,O) = S(0)-1/2. Returning to (34) we see that if a(O) = wing conclusions :
° and s(o) > °we have the follo-
°
(i) for all A near with a(A) > 0, (34) has no nontrivial solutions near z = 0; for such A there are only axisymmetric solutions in a neighbourhood of the origin;
°
(ii) for all A near with a(A) < 0, (34) has a small "circle" of solutions near z = 0, given in polar coordinates by
P
=
PA(e)
=
la(A) 11/2n* (e,A, lap) 11/2) ;
one has k
= 0,1,2
°
and PAce) .... as a(A) .... ,0; these solutions generate a family of nonaxisymmetric solutions of the original problem (1). We conclude that as we change A in a neighbourhood of the origin, such that a(A) goes from positive to negative, then there is a family of non-axisymmetric solutions of (1) bifurcating from the trivial solution.
260
7 Bifurcation problems with SO(2)symmetry and Hopf bifurcation 7.1. INTRODUCTION In the recent past much research has been devoted to the study of the Hopf bifurcation phenomenon, that is the bifurcation of periodic solutions from equilibrium solutions for autonomous differential equations. As a general reference we can refer to the book by Marsden and McCracken [156], which also contains an extensive bibliography. The original contribution by E. Hopf [98] considered one parameter families of autonomous ordinary differential equations, and studied the local bifurcation of periodic solutions from an equilibrium at the origin; such bifurcation takes place at parameter values for which there occurs a change in the stability properties of the equilibrium point. Since this first contribution the scope of the theory has been much widened. Several techniques have been used to approach the problem (see e.g. Friedrichs [67], Chafee [34], Schmidt [201], Ruelle [185], Poore [175], Ize [104]); some global bifurcation results have been obtained (Alexander and Yorke [4], Chow, Mallet-Paret and Yorke [44 ],[ 45]); also, Hopf bifurcation has been studied for functional differential equations (Chafee [35], Hale [79],[ 86], Hale and De Oliveira [91 ],[ 170]), partial differential equations (Iooss [100], Joseph and Sattinger [105], Sattinger [192]) and abstract evolution equations (Crandall and Rabinowitz [52], KielhOfer [118], Ize [104]); finally, Golubi tsky and Langford [ 250] used singularity theory to classify perturbations of degenerate Hopf bifurcations. In this chapter we will study Hopf bifurcation as a special case of the general results of chapter 6, section 2. It appears that the general theorY gives us results of the Hopf bifurcation type when we take for G the group SO(2) of rotations in the plane. We will work with spaces of periodic functions, on which SO(2) can be represented by phase shifts. Since the equations under consideration will be autonomous, we obtain a bifurcation problem which is equivariant with respect to SO(2). Our approach has originally been motivated by the work of Chafee [36], which considers the following problem. Suppose that one has an "unperturbed" 261
autonomous ordinary differential equation (1)
which satisfies the following conditions: (i) fO(O) = 0; (ii) ±i are simple eigenvalues of the Jacobian matrix DfO(x); (iii) none of the other eigenvalues is an entire multiple of ±i. These are the usual hypotheses imposed on the unperturbed equation in the classical Hopf bifurcation problem. Now consider the "perturbed equation" x
f(x)
(2)
where f is supposed to be near fO in an appropriate function space. The problem is to determine all periodic solutions of (2) near the origin and with period near 27[. In [36] Chafee shows that such solutions are uniquely determined by the solutions of a scalar equation g(p ,f) = 0 ,
(3)
where g(p,f) is defined and sufficiently smooth for any (p,f) in a small neighbourhood of (O,fO); also g is odd in p. To the solution p = 0 of (3) there corresponds a stationary solution of (2); to each sufficiently small solution p > 0 of (3) there corresponds a family of non-constant periodic solutions of (2); different elements of such family are related to each other by phase shifts. The method also determines the period, which depends on (p,f) and is near to 27[. An earlier version of these results, restricted to the two-dimensional case, can be found in [ 242 1. We will show that these results follow from the general theory of chapter 6; for example, the oddness of g(p,f) in p expresses the equivariance of the bifurcation problem with respect to the group SO(2). In section 2 we start by considering general bifurcation problems which are equivariant with respect to SO(2); we obtain the bifurcation equation and discuss its solutions under a further nondegeneracy condition. In section 3 we make a further analysis of this condition, and show that.it is generically satisfied. Section 4 contains the application on the Hopf bifurcation problem; we give a unified treatment for ordinary as well as functional differential equations. Finally, 262
in section 5 we use some of our results to prove the Liapunov Center Theorem. 7.2. BIFURCATION PROBLEMS WITH SO(2)-SYMMETRY In this section we apply the reduction method of section 6.2 to equations which are equivariant with respect to the group SO(2). Since SO(2) is a major subgroup of 0(2), one might expect some results similar to those of section 6.4. However, since SO(2) contains no reflections, the condition (6.4.6) is no longer satisfied. This gives an additional scalar bifurcation equation, Which is identically satisfied When the problem has 0(2)-symmetry. 7.2.1. The hypotheses. Let X, Z and A be real Banach spaces, and M : n c X x A xJR -7 Z a mapping, defined and of class C2 in a neighbourhood n of the origin in X x A xJR. We will consider the equation M(x,A,a) =
°.
(1)
In view of remark 6.2.9 and the hypotheses which we will make further on, we will assume that : M(O,A,a)
=
°
V(A,a)
(2)
The main hypotheses on M are the following : (H1) M is equivariant with respect to (SO(2),r,r), where r : SO(2) -7 L(X) and SO(2) -7 L(Z) are representations of SO(2) over X, respectively
r:
Z. (H2)
L = DxM(O,O,O) is a Fredholm operator with zero index; dim ker L and ker L has a basis {u 1 ,u 2 } such that: r(a)u 1 = cosa.u 1 - sina.u Z r(a)u 2 = sina.u 1
CH3)
+
Va
E
JR •
2,
(3)
cosa.u2 '
D D M(O,O,O).u ¢ R(L) for some u a x
E
ker L\ {a} •
263
7.Z.Z. Remarks. It follows from theorem Z.6.17 that, if the representation induced by r on ker L is irreducible, then either dim ker L 1 and r(a)u = u, Va E JR, Vu E ker L, or dim ker L = Z and ker L has a basis {u 1 ,u Z} such that: coska.u 1
sinka.u Z
Va
E
JR ,
(4)
sinka.u 1 + coska.u Z
for some k E N\ CO}. The hypothesis (HZ) rules out the case dim ker L = 1, which can be handled as in section 6.3; all bifurcating solutions will in this case satisfy r(a)x = x, Va EJR. (HZ) says that dim ker L = Z and that we have k = 1 in (4); the following argwnent shows that this gives no loss of generality. Assume that (H1)-(H3) are satisfied, with (3) replaced by (4), for some general k f O. Then Vu E ker L , P
=
O,l, ... ,k-l .
(5)
It follows from section 3.4 that each solution (X,A,O) of (1) sufficiently near the origin will have the same invariance property as the elements of ker L : Z'[[ ,
r (TP)X = x
P
O,l, .•. ,k-l.
(6)
Define Xk
{xEX\ fCt'P)X=x, p=O,l, ... ,k-1}
and ~ Z'[[ Zk = {zEZ \ r(l,(P)z=z, p=O,l, ... ,k-l}
Let ~
be the restriction of M to dom ~ = ~ = ~ n eXk x J\ xJR); then into Zk' and our problem (1) is equivalent to the equation: ~
~(X,A,O) =
Z64
°.
~
maps
(7)
We will show that (7) satisfies (H1)-(H3). Define representations fk : SO(2) ~ L(Xk ) and fk follows : Va
SO(2) ~ L(Zk) as
E
JR •
(8)
It is clear that Mk is equivariant with respect to (SO(2),fk ,fk ), so that (7) satisfies (H1). Let Lk = DxMk(O,O,O); then, since ker L C Xk by (4), we have ker Lk = ker L, and (3) is satisfied if we replace f and f by fk and f k . Also, R(~) = R(L) n;. Indeed, it is clear that R(~) C R(L) n Zk' since Lk = Llx and since L cOIIllIRltes with f(a). Conversely, if z E R(L) nz k , then z = Lx ~r some x E X, and, since z E ; , we have : 1 k-1 2n 1 k-1 - 2n z = k I f(l(p)z = L(k I f(l(p)x) , p=o p=O 1 k-1 2n where k I f(l(p)x E xk ; so z E R(~). If Q E L(Z) is a projection such that p=O ker Q = R(L) and f(a)Q = Qf(a), Va E JR, then lerrnna 6.2.4 shows that R(Q) C Zk' Then it follows from R(~) = R(L)nZ k = {zEZkl Qz=O} that ~ is a Fredholm operator with zero index. Finally (7) also satisfies (H3), as can easily be seen from the foregoing and DaJCk D M (0,0,0) = Dax D M(O,O,O) Ixk . We conclude that there is no loss of generality in restricting to the case k = 1. In the application on differential equations given in section 4, the foregoing reduction can be performed by a simple time rescale. 7.2.3. It is clear that the hypotheses (H1)-(H3) above form a particular case of the general hypotheses (H1)-(H3) of section 6.2, so that we can apply the results of that section, and in particular theorem 6.2.17. Also lerrnna 6.2.16 applies, since (3) shows that f(n)u = -u for each u E ker L. By theorem " 6.2.17 we have to find solutions (p,e,A) of the bifurcation equation
G(p,e,A)
=
°,
(9)
°
where (p,A) are near the origin, p > and e E S = {i31uti32U2 I i3~+i3~ = 1}. The mapping G is of class C1 , and satisfies (6.2.47)-(6.2.50) and (6.2.52). 265
Since each e E S has the form e = r(a)u 1 for some a E IR, and G(p,r(a)u 1 ,A) r(a)G(p,u 1 ,A), it follows that (p,e,A) solves (9) if and only if (p,A) is a solution of G(p,U 1 ,A) = 0
(10)
Since 0 and g(p,A) = O}
The mappings x, ~ and g have the following properties, for all (p,A) E w (i)
X(O,A) = 0 , rX(p,A) X(-p,A)
(ii)
~(O,O)
r( rr)x(p ,A)
( 12)
0 and
~( -p,A) = ~(p,A)
266
pu 1 , and
(13)
(iii) g(O,O) and g(-p,A)
= g(p,A)
o
(14)
7.2.5. For further use, let us recall here how the mapping obtained. Let Q E L(Z) be a projection such that ker Q r(a)Q, Va
E
R. Let v* (u,A,a) be the unique solution v
(I-Q)M(u+v,A,a)
=0
i, cr
= R(L)
E
and g can be
and Qf(a)
.=
ker P of the equation
.
(15)
F(u,A,a) = B-1 QM(u+V*(u,A,a),A,a)
(16)
Then (17) where the scalar functions fi (p,A,a) (i = 1,2) can also be obtained as follows. Define Wi
E
R(Q) (i = 1,2) by
w. = QD D M(O,O,O).u. 1
a x
1
Then there are functionals Qi
(i =1,2) .
(18)
Z +R such that Vz E Z •
(19)
The functions f.(p,A,a) are given by 1
i
1,2
(20)
they are odd in p, and can be written in the forn i = 1,2 ,
(21)
where 267
1
h.(p,A,a) = f
D f.(tp,A,a)dt
lOP
i
=
1,2
1
(2Z)
The function O(p,A) is the unique solution of the equation h 1 (p,A,a)
o,
(Z3)
while g(p,A) = hz(p,A,o(p,A))
(Z4)
Finally, X(p,A) is given by X(p,A)
= pU 1 +
v*(pu 1 ,A,o(p,A))
(Z5)
7.2.6. SO(Z)-symmetry versus O(Z)-symmetry. Suppose that the equation (1) is equivariant with respect to some representation of the group 0(2), while also (HZ) and (H3) are satisfied. (This is the situation studied in section 6.4). Then of course the foregoing results remain valid. But now one can choose {u 1 ,u Z} such that r Lu 1 = u 1 and rLu Z = -u Z' and in addition to the foregoing properties we also have r LF(u,A,a) = F(r Lu,A,a). This implies that F(pu 1 ,A,a) = F(pr Lu 1 ,A,a) = r LF(pu 1 ,A,a), and consequently f 2 (p,A,a) = 0 for all (p,A,a). Then we see from (Z4) that also g(p,A) = 0 for all (p,A), i.e. the bifurcation equation (11) is identically satisfied. As a consequence, (X(p,A),A,O(p,A)) will be a solution of (1) for ~b~~y (p,A) E w. In case M is only equivariant with respect to SO(2), then this argument fails, and there remains the bifurcation equation (11) to solve: (X(p,A),A, O(p,A)) will only be a solution of (1) if (p,A) satisfies (11). In his recent book [2471, Sattinger came to a similar conclusion using a rather different approach. Now there are different ways to attack the equation (11). We consider first the classical approach, which consists in solving (11) for A as a function of p; of course, in order to be able to do so, we have to restrict to one single scalar parameter A E JR.
Z68
7.2.7. Theorem. Let M satisfy (2) and (H1) - (H3), with A
= JR.
Assume that (26)
Then there is a neighbourhood r21 of the origin in X xffi. xffi., a Po > 0 and C1_ mappings x: ]-PO,PO[ +X, ~: ]-PO,PO[ +JR and 0 : J-PO'PO[ ->-JRsuch that the nontrivial solutions (X,A,O) E r21 of (1) are given by : {(r(a}x(p),5:.(p),O(p)) Furthermore x(O)
0, ~(O)
I aEJR,
O.'0,° 0 ) in 1\ xJR, and L : w1 + L(X,Z) a C -map. Assume the following: 274
(H)L (i)
(ii)
L(A,a)r(a) = f(a)L(A,a)
,
Va
E
JR , V(A,a)
E
w1 ;
(9)
LO = L(AO,aO) is a Fredholm operator with zero index, dim ker LO 2, and ker LO has a basis {u 1,u2} such that (4) holds;
We wish to determine ker L(A,a) and R(L(A,a)) for each (A,a) near (AO,a O). 7.3.4. Determination of ker L(A,a) and R(L(A,a)). Consider the linear equation L(A,J).x = z , where z E Z is be projections both commuting E ker LO and v
(10)
given, and we want to solve for x E X. As usual, let P and Q onto ker LO and onto a complement of R(LO) , respectively, with the synnnetry operators. Write x E X as u+v, with u = Px = (I-P)x E ker P. Consider the equation:
(I-Q)L(A,a).(u+v)
(I-Q)z
(11)
Since LO = (I-Q)L(AO,aO) is an isomorphism between ker P and R(LO) , the same holds for (I-Q)L(A,a), if (A,a) is sufficiently close to (AO,a O). So we can solve (11) for v; we obtain v = V*(A,a).u + W*(A,a).z ,
(12)
where for each (A,a) near (AO,ao) , we have V*(A,a) E L(ker LO,ker P) and W*(A,a) E L(Z,ker P). Also V*(AO,a O) = 0, and V*(A,a)r(a) = r(a)V*(A,a)
, W*(A,a)f(a)
r(a)W*(A,a) .
(13)
Equation (10) reduces to QL(A,a).(u+V*(A,a).u) = Q(A,a).z ,
(14)
275
where Q(A,O).Z = QZ - QL(A,O).W*(A,O').Z
Vz
E
Z •
(15)
Since W*(A,O').Z has the form W*(A,o).(I-Q)z, it is easily seen that Q(A,O') is a projection in Z onto R(Q) and connrruting with the 'rCa). Using the condition CH)L(iii) the argument of lemma 6.2.4 shows that induces on R(Q) an irreducible representation of SO(2) equivalent to the representation (4). An appropriate basis for R(Q) is given by {w 1,w2}, where
r
wi = QDoL(AO'O'O) .ui (i = 1,2). Consider now the operator C(A,O') E L(ker LO,R(Q)) defined by C(A,O) = QL(A,O')(Ik
L +V*(A,O')) . er 0
(16)
Since C(A,O')r(a) = 'r(a)C(A,O'), Va EIR, it follows from Schur's lemma (see section 2.6) that either C(A,O') is an isomorphism, or C(A,O') = O. In case C(A,O) is an isomorphism equation (14) has a unique solution u for each z E Z; in particular, taking z = 0, we see that ker LeA,O') = {O}. In case C(A,O') = 0 (14) (with z = 0) shows that ker LeA,O') = {U+V*(A,O').U I uE ker LO}; since the mapping u O. One has pem) = O(ly(m) 11/2). The set BO = {mEU 1 y(m) =O}
(30)
r-1
is' a C -submanifold of AD, with codimension one, contaInIng ~, and dividing U in two subregions : in one of these subregions (1) has no nontrivial solutions in w, while for m in the other subregion (1) has exactly one family of nontrivial solutions as described above. Proof. Apply theorem 2.9. The fact that BO is a submanifold of codimension one was already proved in theorem 7. The fact that x(m) = O( 1y(m) 11/2) shows that BO is indeed the (local) bifurcation set of the problem : if m approaches BO from the subregion where (29) is satisfied, then the corresponding family of nontrival solutions approaches x = o. 0 7.3.11. Remark. When considering a one-parameter family of equations (A E JR), such as in theorem 2.7, then we consider in fact a one-dimensional path in
AO through mO("') = M(.,O,.). The condition (2.26) means that this path is tAan6v~~aL to the submanifold BO at mO (see (2.28)) : the path crosses the bifurcation manifold BO' and bifurcation takes place. In the paper [268] we briefly discussed a situation where the condition a f 0 is not satisfied. This (nongeneric) situation is interesting from the point of view of imperfect bifurcation (see Golubitsky and Schaeffer [74] and [75], and Golubitsky and Langford [250]). 7.4. HOPF BIFURCATION In this section we consider a particular application of the results of section 2. We suppose that M is a mapping of a specified form between spaces of periodic functions. As a special case we obtain the classical Hopf bifurcation for ordinary differential equations as well as for functional differential equations. Our general setting gives a unified approach for both cases.
281
7.4.1. The problem. Let Z be the space of all continuous 2n-periodic functions Z : R ~ ill:n , let X be the subspace of all continuously differentiable functions in Z, and let M(X,A,O) = - ~ + F(X,A,O) . Here x E X, A E A and a map such that F(O,A,O)
E
(1)
JR, while we suppose that F
o
V(A,O)
The equation M(X,A,O)
o takes
Z x A xJR ~ Z is a C2_
(2)
A xR •
E
the form
dx
dt = F(X,A,O) .
(3)
This is a functional differential equation for the 2n-periodic function x = x(t). First we examine what conditions should be imposed on F such that M is equivariant with respect to SO(2). This leads to the concept of an autonomo~ equation, as follows. 7.4.2. The symmetry condition. There is an obvious representation of SO(2) over Z, defined by (r(a)z)(t) = z(t+a)
Vt
E
R , Va
E
R , Vz
Z •
E
(4)
The restriction of rea) to X gives a corresponding representation over X. The mapping M defined by (1) will be equivariant with respect to (SO(2),r) (i.e. M(r(a)x,A,o) r(a)M(x~A,O), Va E R, V(X,A,O) E X x A xJR) i f and only if F(r(a)z,A,o)
= r(a)F(z,A,O)
Va
E
JR , V(Z,A,O)
E
Z xA xJR •
(5)
This follows from the continuity of F and the fact that X is dense in Z. Condition (5) says that equation (3) is autonomo~, that is invariant for time translations. An equivalent form for the condition (5) is
282
F(Z,A,O)(t)
= F(r(t)Z,A,O)(O)
Vt E JR, V(Z,A,O) E Z x A xJR .
When this is satisfied, let us define f : Z x p, xJR f(z,A,o)
=
->- JRn
(6)
by
F(Z,A,O)(O) .
(7)
We also denote r(t)z by Zt' i.e. (r(t)z)(s)
=
z(t+s)
Vs, t E JR , Vz E Z •
(8)
So Zt(s) is 2n-periodic, both in s and t. This notation has been inspired by a similar notation introduced by Hale in the theory of functional differential equations (see e.g. [79]). However, its meaning here is slightly different, since we restrict to periodic functions, and allow s to take values in all JR (and not only in an interval on the negative axis).
Using (6) and the notation (8) we can rewrite (3) in the form Vt
E
JR .
(9)
M is equivariant with respect to (SO(2),r) if and only if (3) can be rewrit-
ten in the form (9), for some C2-map f : Z x A xJR
->-JIf.
In order to translate the hypotheses (H2) and (H3) of section 2 in terms of our particular M, we have to study the linearization of M. 7.4.3. The linearization. Fix some (AO'OO) E AxJR. We define, for each
0
EJR:
(10)
Vx EX, Vt E JR
(11 )
We want to determine ker L and R(L). To do so we have to study the linear equation
z (t)
Vt
E
JR ,
(12) 283
where z E Z is given, and we want to solve for x E X. We will use Fourier series to detennine the solutions of (12). For each ~[Z]
k E I we denote by ~[ z
1=
1 J2Tr
2 Tr
0
e
the k-th Fourier coefficient of z E Z :
-ikt
.
z (t) dt
;
( 13)
since z is real valued, we have c_ k[ z)
~[Z). For each N E IN we denote by
PNz the trwlcated Fourier series of z : (14 )
On Z , the complexification of Z, we use the bilinear fonn c
1 J2Tr = 2Tr 0 (z* (t),z(t))dt
Vz ,z* E Z ,
where (.,.) is the usual inner product on ~ : (a,b)
(15) n
=
I
a.b., Va,b E (Cn. I
i=l
I
We need the following facts about the Fourier expansion of z. E Z : we have I IIck[ z )11 2 < co, and PNz converges in L2 ((O,2Tr) ,lRn) to z; consequently, we kEI have for each z,z* E Z : lim N->w
= .
If x E X, then ck[x)
= ikck[x), and
(16 )
I
(1+k2)IICk[X]iI2
kO the operator ilk is invertible, nk = 0 and II ilk 111 ~ CIkl- 1 for some C > O. Now notice that {Re(cke ikt ) I ck E ker ilk} is a 2nk -dimensional (real) subspace of X if k f 0, and a nO-dimensional subspace if k = o. Then, using 285
n_k = nk , (21) is an easy consequence of (22). So it remains to prove (22). Let x E ker L. Taking z =·0 in (18), we see that ~[X] E ker ~ for all k E~, and consequently ck[x] = 0 for Ikl > kO. But then: x(t) =
\ ikt L ~[x]e , Ikl.;;;k O
with ck[x] E ker llk and c_k[x] = ~[x], so that x belOIlgs to the set at the r.h.s. of (22). Conversely, a direct calculation shows that the function Re(cke ikt ) belongs to ker L if ~ E ker"llk. This completes the proof. 0 7.4.5. Lemma. R(L) is closed and dim ker L = codim R(L) , i.e. L is a Fredholm operator with zero index. Also
llk
Proof. Since dim ker = dim ker llk = ~, the first part of the lemma is a consequence of (23) and lemma 4. So it is sufficient to prove (23). Let z E R(L); then (12) and (18) hold for some x E X, i.e. ck[ z] E R(~) for all k E~. This is equivalent to : (24)
Ck
llk
Since ker ~k is trivial for Ikl > kO' and since E ker if and oply if E ker ll:k' it follows that z belongs to the set at the r.h.s. of (23). Conversely, if z is an element of the set at the r.h.s. of (23), then z E Z and (24) holds. This implies that the equation
2k
has at least one solution ck E ~ for each k E~. If ~ is a solution of (25)k' then ~ is a solution of (25) -k. Let now {~ I kE~} be any. solution set of (25), such that c_ k = k • Consider the Fourier series I~elkt; we will show that this series converges to some x E X, and that Lx = z; so Z E R(L) , and the lemma will be proved.
c
If Ikl
> kO' then (23)k and lemma 4 imply that ck
Clkl-1I1Ck[ z]1I. So we have: 286
=
-1
llk .~[ z] and II ~II
N-roo t=O N-roo ,t=2n (\jJ(t) ,x(t)) I -lim I t=O N-roo j t=2n - = (\)J(t) ,y(t)) I t=O
(t]1
-
By a standard argument this implies that x(t) = yet) for all t
ElR.
0
7.4.6. We want to find conditions on f(z,A,o) such that M(Z,A,O) satisfies the hypotheses (HZ) and (H3) of section 2. To formulate such conditions we will need the operator I'Ik(o) Dk(o).c = -ikc
+
E
A(o)(ce
n
L(CC: ) defined by :
iks
)
Vc E
ccf1
(27)
where A(o) is given by (10). We denote by Dk(O) the derivative of Dk(o) in the variable o. Also, let (28)
Finally, we remind the definition 6.3.1 of a B-simple eigenvalue of a linear 287
operator A.
.
Z
7.4.7. Theorem. Let (A O,aO) E A xlR, and let f : Z x A xlR -+:nf be a C -map such that f(O,A,a) = 0, V(A,a) E AxlR. Let M be defined by (1), where F : ZxAxlR -+ Z is defined by F(z,A,a)(t) = f(zt,A,a). Let L, A(a), ~k(a) and Hk(a) be given by ('1), (10), (Z7) and (Z8) respectively. Then the following statements are equivalent : (i)
dim ker L
Z, ker L has a basis {u1 ,uZ} such that (Z.3) holds, and DaD~(O,AO,aO).u ¢ R(L) for some u E ker L\ {a};
(ii)
zero is a ~;(aO)-simple eigenvalue of for all k 1= ±1;
=
~l(aO)'
and
~k(aO)
is invertible
(iii) aO is a simple zero of Hl (i.e. H1 (aO) = 0 and H;(a O) 1= 0), and Hk(aO) 1= 0 for all k 1= ±1. P r.o 0 f. Suppose that (i) is satisfied. Let c 1 = u 1 (0)+iu Z(O), and c;(t) = c 1e I t • It follows from (Z.3) that u,(t) = Re(C;(t))1 uZ(t) = 1m (C;(t)) and ker L = {Re(zC;) I zECC}. If u E ker L, then ck[ ul = 0 for all k 1= ±1. By lemma 4 this implies that nk = 0 for k 1= ±1, and that ~k = ~(aO) is invertible for such k. Since dim ker L = Z, (Zl) also shows that n, = n_l = 1. Indeed, we have ker ~l(aO) = span{c 1}· Now, by definition of ~l(a) and c 1 :
and
Since this last expression does not belong to R(L) (by (i) and the general theory), it follows from lemma 5 that (ci'~1(aO).cl) 1= 0, where ci E cc;ll is such that ker ~i(ao) = span{ci}· But this means precisely that ~1(aO).c, ¢ R(~l(aO))' i.e. zero is a ~;(aO)-simple eigenvalue of ~l(aO). Conversely, suppose that (ii) holds. Then lemma 4 shows that dim ker L = 2 and ker L = {Re(zC;) I zECC}, where c;et) = c 1eit and c 1 is defined by Z88
ker 6,(° 0 ) = span{c,}. It is clear that the basis {u, ,u 2 } of ker L, given by u, = Re sand u 2 = 1m s has the required transformation properties. Since also (Ci,6,(00)'c,) of 0, the calculations above show that DoDxM(O,AO'OO)'u, ct R(L). We conclude that (i) and (ii) are equivalent. As for the equivalence of (ii) and (iii), it is clear that 6k (00) will be
invertible if and only if Hk(oO) of O. The remaining part of the proof follows from the following general result. 0 7.4.8. Lemma. Let 6 : JR+ L(Cn) be defined and of class C' in a neighbourhood of ° = 00' Let H(o) = det 6(0). Then zero is a 6'(00)-simple eigenvalue of 6(00) if and only if 00 is a simple zero of H, i.e. if and only if H(oO) = 0 and H' (° 0 ) of
o.
Proof. A proof using the canonical Jordan form of 6(00) can be found in Hale and de Oliveira [9'],[ 170]. Here we give a proof based on exterior forms (see [238]). We start with the following identity, which can be considered as the definition of H(o)
Vo E JR ,
(29)
where {c" ... ,cn } is any basis of (Cn. Suppose zero is a 6'(00)-simple eigenvalue of 6(00)' Choose the basis of ten such that 6(00)'c, = O. Then {6(00).cj I j =2,3, ... ,n} will form a basis
°
for R(6(00))' Putting = 00 in (29) gives H(oO) = 0, since c,/\ ... "cn f O. Differentiation of (29) at = 0 gives then
° °
(30) Since 6' (°0 ) .c, 'I- R(L), the right hand side of (30) is different from zero, and we conclude that H' (00) of O.
°
Conversely, suppose that H(oO) = and Ii' (00) f O. Then (29), for ° = 00 and any basis of ten, shows that the vectors {6(00).cj I j = ', ... ,n} are linearly dependent, and consequently dim ker 6(00) = k? ,. Let now {c" ... ,cn } be
°
a basis of a;:ll such that 6(00) .c j = for , ~ j ~ k. Then (30) holds, and k> " it follows that H'(oO) = 0, which contradicts the hypothesis. So
if
289
d:im ker 1l(00) = 1. Furthermore, (30) and H' (° 0) f 0 :imply that ll' (° 0) .c 1 does not belong to span{ll(00).c j l j =2, ... ,n} = R(ll(00))' So zero is a ll'(oO)simple eigenvalue of 1l(00)' 0 7.4.9. The projection operators. When the equivalent conditions (i), (ii) and (iii) of theorem 7 are satisfied, then we can apply the general results of section 2 to the equation (9). In order to find a more explicit expression for the bifurcation function g(O,A) we need to introduce appropriate projections P and Q. Fix c 1 E ker 111 (00) and ci E ker lli (° 0) such that
it it Let C;(t) '" c 1e ,C;* (t) = c e ,u l (t) '" Re C;(t) and u 2 (t) = 1m c;(t) " Then ker L = span{u 1 ,u 2} '" {Re(zc;) l zECC} and R(L) = {zEZ I =O}, as can easily be seen from lemma 4 and lemma 5. Define PEL (X) by :
1
Vx E' X
(32)
From this it follows easily that P is a projection onto ker L, such that pr(a) = r(a)P, for all a E R. Similarly, we define Q E L(Z) by Qz = Re«c;* ,z>X)
Vz E Z ,
(33)
where X(t) = 1l;(00).c 1e it = DoA(oO).C;t" Again, Q is a projection commuting with the symmetry operators, and such that ker Q = R(L). If we let w.1 290
QDoD}1(O,A O,oO)·u i = QDOA(OO).ui,t (i=1,2), thenw 1 = Re(X) andw2 From this it follows that : Q1(z) = Re 0, then yet) = x(.!) is a 2no-periodic solution of (36). So we have to determine 2no periodic solutions of (39). To bring this in the fonn (9), define f : Z x fI. x 1 0,00 [ -+ JRn by :
(40)
f(z,A,o) = oh(zO ,0 ,A) . Now remark that for each s E [-r,O] (Zt)O ,0 (s) = (r(t)z)O ,0 (s) = z(t+~) 0
Zt ,0 (s) .
It follows that (9), with f defined by (40), coincides with the equation (39). Remark in this context that 0 has to be considered as an unknown of the problem : given A E fI. one has to adjust 0 such that (39) has a 2n-periodic solution; if such 0 can be found, then (36) has a corresponding 2no-periodic solution. For the critical value (AO'oO) we will take 0 0 = 1; the general case can be reduced to this particular one by a preliminary time rescale in the equation (36), and an appropriate redefinition of the mapping h. So we will look for solutions (x,A,o) E Xxfl.x]O,oo[ of (39) near (O,A O,l). 7.4.11. Remark. The mapping f defined by (40) is only continuous in
but not continuously differentiable : this is due to the presence of the tenn Zo . But when we consider the restriction of f to X x fI. x] 0,00 [, then f is of cl~~s C1 in 0, and of class C2 in (x,A). This is sufficient for the theory of the previous sections to remain valid. 292
0,
In a number of cases a similar remark can be made about differentiability in the parameter A. For example, equation (36) may contain some delays; considering these delays as parameters, f will only be continuous in these parameters. Again, the restriction of f to X will be continuously differentiable (see IIale [ 86]). A general remark in this context is that when we differentiate in A and 0, this usually only happens after replacing x by solutions of certain equations, and such solutions may have more smoothness than general x. ~ow define A
~.
L (C ,lRn), 6 : II.
-7
x
(C
-7
L((Cn) and H : II. x (C
-7
(C by :
A(A) = Dzh(O,A) , 6(A,~).C
=
(41 )
-~c + A(A).(ce~s)
Vc E
(Cn ,
(42)
and det
6(A,~)
.
( 43)
is easily seen that these mappings are of class C1 ; in particular, H(A,~) is analytic in)1. We let II' CA,~) = D !-I(A,~) and 6' (A,~) = D 6(A,~).
It
~
~
7.4.12. Lemma. For the equation (39) and 0 0 = 1, the statements (i), (ii) and (iii) of theorem 7 are equivalent with : (iv)
HCAO,i) = 0 , H' (AO,i) f 0 and H(AO,ik) f 0 , for all k E ~ , k f ±1.
Proof. From the definition (27) of 6k (0) we find for the case under consideration : -ikc + OA(AO)·(ce -ikc
+
OA(AO).(ce
iks
)0,0
iks/o
) Vc E
e .
Consequently
293
Vo > 0 , VI
=
-
Vz E Z •
(60)
,z> =
Vz E Z •
(61)
'"
and Q2(z)
= -Im- R(Q) is continuous, F(s ,0)
~ -1 F(S.SO,A) = r(sO )F(S,A)
=
0, Vs E G, and
Vs o EGO' Vs E G, VA E w .
(12)
8.2.11. Theorem. Assume (Hl), (H2) and (H3) , and let Q E L(Z) be a projection satisfying (7). Then there exist a neighbourhood U of YO in X, and a neighbourhood w of the origin in A such that each solution (X,A) E U x w of (1) can be written in the form x
=
r (s )( Xo +y* (s , A) ) ,
(13)
where (S,A) is a solution of (11) and Y*(S,A) is the mapping given by lemma 10. Conversely, if (S,A) E Gxw is a solution of (11), and x is given by (13), then (X,A) is a solution of (1). The representation (13) of solutions of (1) in U x w is unique modulo right multiplication by elements of GO; that means, two solutions (S,A) and (t,A) of (11) give the same solution of (1) if and only ift = s.sO for some So E GO.
0
This theorem reduces the problem (1) to that of solving the bifurcation equation (11). For a discussion of some aspects of this equation, one can see Vanderbauwhede [ 230]. In section 5 we will discuss a particular example of such bifurcation equation. 8.3. SYMMETRIC SOLUTIONS In this section we show how the symmetry properties of the equations may sometimes imply the existence of certain branches of solutions near a compact manifold of solutions of the unperturbed equation. Our presentation is independent from the results of the preceding section, and is based on the ideas in Vanderbauwhede [ 231] and on a reformulation of these ideas by D. Chillingworth (private communication). Dancer [ 248] gives a somewhat different approach. 8.3.1. The hypotheses. Let X, Z and A be real Banach spaces, and M : Xx A-+ Z a continuous mapping which is continuously differentiable in x E X. Consider 315
the equation M(x, A)
o.
(1)
We make the following hypothesis :
(H) There is a compact group G, a closed subgroup H, and representations r : G + L(X) and G + L(Z) such that :
r:
M(r(S)X,A) = r(s)M(x,A)
Vs E H , V(X,A) E X x A
(2)
Vs E G , Vx EX; .
(3)
and M(r(s)x,o) = r(s)M(x,O)
i.e. Mis equivariant with respect to (H,r,r), and for A
0 even equi-
variant with respect to G. We will also use the following notation
if T
C
G, then we let
FixX(T) = {xEX I r(s)x=x, VsET} .
(4)
8.3.2. Theorem. Assume (H), and let Xo E X be such that (i) (ii)
M(xO'O) = 0 ; L = D~(XO'O) is a Fredholm operator
(iii) Fixz(H O)
C R(L) ,
where HO = {sEH I r(s)xO =xO} is the isotropy subgroup of xO· Then there exists a mapping : A +> x* (A)
defined and continuous for A in a neighbourhood w of the origin in A, and with x* (0) = xo ' such that M(x* (A) ,A) 316
o
VA
E w •
(5)
Proof. Since Lr(s) =r(s)L, Vs EGO = {tEGI r(t)xO=x O}' there exist projections P E L(X) and Q E L(Z) such that R(P) = ker L
ker Q = R(L) ,
pres) = r(s)p
Qr (s) =
and
Now define F : ker P x A
-+
r (s) Q
R(L) by (6)
F(y,A) = (I-Q)M(xO+y,A) .
We have F(O,O) = 0, while Dy F(O,O) = Llk er P is an isomorphism between ker P and R(L). It follows that the equation F(y,A) = 0 has a unique solution y = y* (A), with y*
: we 1\
-+
ker P continuous and y* (0) = O.
Since F(r(S)y,A) = r(s)F(y,A), for all s E HO' and all (y,A) E ker pxl\, it follows that r(s)y* (A) = y* (A), Vs E HO' VA E w, i.e. y* (A) E FixX(H O), VA E w. This in turn implies that VA E w • Defining x*
: w
-+
X by x* (A) = xO+Y* (A), (5) then follows from Fey* (A) ,A) = 0,
VA E w. It is also clear that x* (A) E Fixx(H o)' VA E w. 8.3.3. Let now
Xo
0
E X satisfy the conditions (i) and (ii) of theorem 2. Then
the same conditions are also satisfied at each point ~O of the orbit YO = ~ -1 {r(t)x o 1 tE G}, since we have DxM(r(t)xO'O) = r(t)DxM(xO,O)r(t ). So we may ask at which points of YO condition (iii) of theorem 2 is satisfied. Let ~O = r(t)x O (tE G), and let
be the isotropy subgroup of ~O. Then -1 it is not too difficult to see that HO = H n t.G O. t ,and consequently
Ho
~
Since R(DxM(~O'O)) = r(t)R(L), we have the following result.
317
8.3.4. Corollary. Assume (H), and let Xo E X satisfy the conditions (i) and
(ii) of theorem 2. Let t E G be such that (iii)'
Fix Z(GO n t
-1
Ht)
C
R(L) .
Then there exists a mapping
defined and continuous in a neighbourhood w of the origin in fl, and with Kt(O)
=
r(t)xO' such that VA
E
o
w •
(7)
8.3.5. Remark. If the condition (iii)' of corollary 4 is satisfied for some
t E G, then it is also satisfied for t'
=
s.t.s O'
~1ere
s E H and So EGO'
We leave it as an exercise to show from the proof of theorem 2 that one will have VA
E
w .
Consequently, all the solutions obtained in this way belong to the orbit of solutions {r (S)Kt (A)
I s E H}
generated by the solution Kt (A).
8.3.6. Remark. Suppose "tv! satisfies the hypotheses (H1)-(H3) of section 2, while also (H) is satisfied. Using the notation of section 2, it is then easily seen that M(s. t,y ,A) = M(t,y ,A), for all 5 E H, t E G and (y,A) E ker P x fl. This implies y*(S.t,A)
y* (t,A)
V(S,t,A} E HxGxw
(8)
V(S,t,A) E HxGxw.
(9)
and P(s. t,A)
F(t,A)
If now s E GOnt
318
-1
Ht, then we have from (9) and (2.12)
- -1 rcs )F(t,X)
= FCt.s,X) = FeCt.s.t -1 ).t,X)
=
F(t,X)
( 10)
1 -1 l.e. F(t,X) E FixZCGOnt- Ht). If also (iii)' holds, then FixzCGOnt Ht) n RCQ) = {OJ, and consequently FCt,!.) = 0 for all ,\ E w. This gives another proof of coronary 4; however, paxt of the hypotheses (HZ) - (H3) are Uill1ccessary to obtain the result, as the direct proof shows. 8.3:7. f~rol1a~. Let X arld Z be real Banach spaces, G a compact group,
r : G
-+
L eX) and
r :G
-f.
L
representations, and
Mo :
X
+
Z a C1-mapping
such that
Vt E G , Vx EX.
(11 )
Assume the following (i)
Xo
E X is such that MO(x O)
operator ; Cii) GO (iii)
=
{tEG I r(t)x O =xO} ;
H is a closed subgroup of G
(iv) t EGis such that
c R(L)
~
(12)
TIlen the equation
(13)
has for each sufficiently small p E Fix,.,(H) an orbit of solutions { (s)x'tCp)
I sEf-{},
where x't
'"'
depends continuously on p, x~(O)
=
r(t)x O'
and
Vp •
Proof. One can apply corollary 4 wi th 1\ defined by M(x,p)
=
tv\J(x)-p.
Fix Z(H) and 1'4
X x 1\
+
Z
0
319
8.4. APPLICATION
PERIODIC PER11JRBATIONS OF CONSERVATIVE SYSTEMS
8.4.1. The problem. In this section we will apply theorem 3.2 and corollary 3.7 to scalar equations of the form:
x+
g(x)
= pet)
(1)
where g : 1R -+ 1R is of class C1 , and P : 1R -+ 1R is continuous and 2rr-periodic, with IIpli = sup{ Ip(t) I I tE1R} sufficiently small. We want to find 2rr-periodic solutions of (1). Let Z = {z : 1R-+1R I z is continuous and 2rr-periodic} and X = {x E Z I x is of class C2}; in Z we use the CO-sup norm, and in X the C2-sup norm. In order to simplify our notation further on, we will identify Z with the set of continuous functions z : Sl -+1R, where Sl is the unit circle in1R2. Define Mo : X -+ Z by MO(x)(t) = x(t)
+
g(x(t))
Vt E 1R , Vx EX.
Then our problem takes the form (3.13). The basic group with which we will work is 0(2) xZ 2 E = ±1}; this group acts on Z (and X) by : r(y,E)Z = EZOy -1
Vz E Z , V(y,E) E 0(2) xl 2 '
(2)
((y,E) I yEO(2),
(3)
(where we identify y E 0(2) with its restriction to Sl). For general g(x) , MO is equivariant with respect to 0(2) x {1} ~ 0(2); if g(x) is odd, then MO is equivariant with respect to 0(2) xZZ8.4.2. The linearization. Let Mo(xo) 2rr-periodic solution of
x+
g(x)
o.
o for
some
Xo
E X, i.e. xO(t) is a
(4)
We will assume that Xo (t) is non-constant, with least period 2rr/k (kEM {On· The operator L = DiMO(xO) E L(X,Z) is explicitly given by
320
(Lx)(t)
=
i(t)
+
g'(xoCt)).x(t)
Vt E]R , Vx EX,
(5)
and has been studied in section 2.2. There we have shown that if hO is the total energy of the solution xO(t), then there exists a function T(h), defined and continuously differentiable for h in a neighbourhood of hO' such that T(h O) = 2njk, while for each h near hO' (4) has a periodic solution with least period T(h) and total energy h. The operator L is a Fredholm operator with zero index, and if (6)
then (7)
and R(L)
=
{zE Z I
Is 1 z(t)xO(t)dt
=
(8)
O}
8.4.3. Lemma. Assume the following: (i)
Xo
is a non-constant 2n-periodic solution of (4), such that (6) holds
(ii) (y,s) E 0(2) xLZ 2 , and y is orientation reversing (iii) xOOy
sxO .
=
Then FixZ(Y's)
C
R(L) .
(9)
Proof. If z E FixZ(Y's), i.e. if zoy change of integration variable that
IS1
z(t)x(t)dt
=
JyeS 1)
sz, then it is easily seen from a
zCt)xO(t)dt
since y is orientation reversing, we conclude that the integral is zero, and consequently z E R(L). D 321
It follows from this lemma that under the condition (6) for
Xo we can
apply theorem 3.Z as soon as .H O contains an element (y,E), with y orientation reversing. An immediate result along these lines is the following. 8.4.4. Theorem. Under the conditions of lemma 3, assume further that g(EX) Eg(X) , Vx E JR.
Then the equation (1) has for each sufficiently small p E Fix Z(y ,E) a Znperiodic solution X' (p) E FiXX(Y,E). Here x* (p) depends continuously on p, and X' (0) = o. Proof. Apply theorem 3.Z, taking for H the (closure of the) subgroup of O(Z) x:2 Z generated by (y,E). 0 8.4.5. Let us now look in a more systematic way for Zn-periodic solutions of (1) bifUrcating from Zn-periodic solutions of (4). Consider first the general case that g(x) is not necessarily odd. Then Mb is equivariant with respect to O(Z). If H is a closed subgroup of O(Z), then (1) will be equivariant with respect to H if and only if p E FixZ(H). Let us restrict ourselves for the moment to subgroups H such that the corresponding class of perturbations FixZ(H) contains functions whose least period is Zn. Then H cannot contain nontrivial pure rotations, and the only nontrivial choice for H is H = {¢(O),oo¢(aO)}' for some fixed 0. 0 EJR (we use the notation of subsection Z.6.13). By a simple phase shift in (1) this can be reduced to tbe case " H = {¢CO) ,o}.
8.4.6. Theorem. Let Xo be a non-constant periodic solution of (4), with least period Zn/k, and such that (6) holds. Then (i) there exists an 0.0 'E JR such that x, (t) = xO(t+aO) and xZ(t) xO(t+aO+n/k) are even periodic solutions of (4) ; (ii) equation (1) has for each sufficiently small p E FixZ(o) (i.e. p = even) at least two different Zn-periodic solutions x;(p) and xZ(p)
(iii) x;(p) and XZ(p) are even fUnctions of t, they depend continuously on p, x;(O) = xl and xz(O) = x Z . 3ZZ
Proof. Let a O EIR be such that xO(aO) = max{xO(t) I tEIR}; then x 1(0) xO(aO) = 0, and since x, (t) solves (4), this implies that x,(t) is even. Since xl (t) is 2n/k-periodic, also x 2 (t) is even. 'This proves (i). Parts (ii) and (iii) follows from an application of theorem 4 to x, and x 2 ' with (y,E) =(0,').
8.4.7.
0
If g(x) in (') is odd, then
Mu
is equivariant with respect to 0(2)
x
'Il2" and H may include elements of the form (y ,-1), y E 0(2). Under the same restriction as before, the nontrivial choices for H become now, after appropriate phase shifts : (i)
H
(ii) H (iii)
H
{(CP(O),l)'Ca,1)}; {C¢(O),l),(a,-l)}; {(¢(O),l),(a,l),(a o ¢(n),-l),(¢(n),-l)}
Case (i) is included in theorem 6. For case Cii) we have the following result, analogous to theorem 6. 8.4.8. Theorem. Let g be odd, and let
Xo
be a periodic solution of (4), with
least period 2n/k, taking both positive and negative values, and such that (6) holds. Then (i) for some So E IR, i,(t)
= xO(t+S O)
and ~2(t)
= xO(t+SO+n/k)
are odd
periodic solutions of (4) (ii) equation (1) has for each sufficiently small p E Fix Z(a,-l) (i.e. p
= odd)
at least two different 2n-periodic solutions ii(p) and
~z(p) (iii) 'X;(p) and xZ(p) are odd functions of t, they depend continuously
2
on p, xi (0) = 'X; and x (0) =
iz .
Proof. One takes So such that xOCS O) (y,E) = (0,-1). 0
= 0,
and one applies theorem 4 with
8.4.9. When we take H equal to the subgroup of O(Z) xZZ generated by (0,1) and (a o ¢(n),-l), then we have to distinguish between k even and k odd. Let
323
Xl be an even periodic solution of (4), with least period 2TI/k, and taking both positive and negative values. If k is odd, then the isotropy group of xl will be the whole H, and we can apply theorem 3.2 at xl and at x 2 (t) = x l (t+l1/k). If k is even, let xj(t) = Xl (t+(j-l)TI/2k), j = 1,2,3,4. Then the isotropy subgroup of xl and x3 will be {(¢(0),1),(0,1)}, while the isotropy subgroup of x 2 and x4 is {(¢(O),l),(oo¢(TI),-l)}. At Xl and x3 we apply theorem 4 with (y,s) = (0,1), at x 2 and x4 we apply the theorem 4 with (y,s) = (oo¢(TI),-l). This gives us the following results. 8.4.10. Theorem. Let g be odd, and Xo as In theorem 8, with k odd. Then (i) there exists an a O E R such that Xl (t) = xO(t+a O) and x 2 (t) = xoCt+aO+TI/k) are even and cxid harmonic pericxiic solutions of (4); Cii) equation (1) has for each sufficiently small p E Z, which is even and odd harmonic, at least two different 2TI-periodic solutions xi (p) and x'2(p) (iii)
Xi (p)
and x'2(p) are even and odd hannonic, they depend continuous-
lyon p, Xi (0) = Xl and xZ(O) = x 2 .
D
8.4.11. Theorem. Let g be odd, and Xo as in theorem 8, with k even. Then (i) there exists an a O E R such that, if we define xj(t) = xO(t+aO+ (j-l)TI/2k), j = 1,2,3,4, then xl and x3 are even, while x 2 and x4 are odd after a phase shift by TI/2 ; (ii) equation (1) has for each sufficiently small p E Z, which is even and odd harmonic, at least four different 2TI-periodic solutions xj (p) (j = 1 ,2,3,4), depending continuously on p and wi th xj (0) x., j = 1,2,3,4. J
(iii) x;(p) and x3(p) are even functions of t, while xZ(p) and x4(p) are odd after a phase shift by TI/2. D 8.4.12. Remarks. (1) Same of the foregoing results C3n also be obtained in a systematic way by using corollary 3.7. When one does not restrict to the subgroup H considered here, this corollary gives a systematic approach to find all symmetry induced solution branches from YO = 324
{r (a)xO
I aEJR},
where
I j I
xo(t) is a 2rr-periodic solution of (4). (2) For the Duffing equation (g(x) = bx+x3), the results of theorems 10 and 11 were proved by Schmidt and Mazzanti [200] using phase plane techniques. (3) As another possible application of the results. of section 3, let us mention the following boundary value problem {l.u + feu) u (x)
=
p(x)
=0
X E
B (10)
X E dB
When the domain B has some symmet ry (e. g. a ball), and when p is small, then one can use the foregoing approach to obtain certain solution near a solution u o of (10) with P = o. 8.5. TIlE BIFURCATION SET
AN EXAMPLE
8.5.1. Introduction. In this section we return to the bifurcation equation (2.11) which resulted from the reduction of section 2. This equation has the form : F(s,:>") = 0
(1)
where F : G x 11 -+ R(Q) is defined ani continuous for (s,:>") near G x {O}, with F(s,O) = 0 for all s E G. If M in (2.1) is of class C1 , then F(s,:>") is continuously differentiable in :>.., and we find from the results of section 2 that: Vs E G •
(2)
By the implicit function theorem (1) can only have nontrivial solutions (i.e. solutions with:>" f 0) near those points (s,O) where D:>..F(S,O) is not injective. If dim 11 ~ dim R(Q) , then generically D:>..F(s,O) will be injective, and in general nontrivial solutions of (1) will only branch off G x {OJ at isolated points. When to the contrary dim 11 > dim R(Q) then D:>..F(s,O) will have a nontrivial kernel for all s E G, and one expects (1) to have nontrivial solutions for all s E G. In [ 230] we have given a general discussion of 325
i the solution set and the bifurcation points for this case. Since this discussion is rather abstract, we wi 11 explain heTe the essential ideas on particular example, which is suffici enUy simple to
allm~ fOT
2
an easy visua-
lization of what is going on. 8.5.2. cnl~-E_:roblem. Let g : lR h
-7
lR and h : JRxJR
2
ry
->
JR be of class C-\ with
,x'Y''\1 ''\2) hT-periodic in the first variable t, and h(t,x,y,O,O) . We want to study 21T-perLodic solutions of the equation
~
0 for
all (t,x
i
+
o
,x,i,'\)
g(x) + h
(3)
I
for ,\ near the oTigin inlR". Special cases of this problem have been studied by Hale and Taboas ([ 81 J ,[ 89 D. ]11e smoothness conditions 'on g and hare such that at each step of the discussion below He v,ill have automatically all the smoothness we need; by a more careEuE analysis it may be possible to obtain the results under weaker smoothness conch bans. To bring the problem in the fonn (2.1) we use the same spaces X and Z as ]
in section 4, take p, '"
JR~
x(t) + g
?>l(x,'\) (t)
and define !'vI : X x J\ (t))+h(t,x(t)
-~
,xCt) ,,\) ,
(4 ) )
Vt E JR , Vx EX, V,\ E
° the equation
(3) becomes
o
i + g(x)
M(x,O) is equivariant with respect to G tion U(cx)zCt)
m.~
=
so (2), wi th the usual representa-
zCt+cx)).
, wjtn least period
Let xoCt) be a non-constant periodic solution of 21r/k (k EN), and such that (4.6) holds. Then L '"
,0) js a Fredholm
opeTatoT, wi th ker Land ReL) given by (Ll. 7), respectively projections P and Q we take P
Qz and 326
,
.8). For the
= Q:X' with
211 -1· . a O x o Jr z(t)xO(t)dt O
Vz
E
Z ,
(6)
Ii
(7)
Since xO(t) is a solution of (5),
Xo
is o[ class C5 , and from this it
easily follows that all hypotheses of section 2 aTe satisfied. The operator
Mdefined
(2.
takes the £on11 : (t)+y(t)) (8) 7
Vt
and is clearly of class C3
E
ill , Va E lR , VCy,,\) c:: ker P xIR'"
Consequently, also the solution y* (a,;,) of the
auxiliary equation (2.8a) and the bifurcation function Fea,A) will be of class C3 . Since dim R(Q) :;: 1 lve may consider F to be real-valued) and explicitly given by
,A)
r2
'IT
(9)
JO Mea.,y'
r (21T.) I· . follows from (2.12) that 5 ( o:,A) will be Also, since GO = lQl(-J . J El } , It
21T/k-periodic in a. Solutions (a,A) and (a
+?£~j ,A) of the bifurcation equa-
tion
Fea,A)
o
(10)
correspond to the same solution of the original problem. 8.5.3. The solution set. In orcler to find the solution set of (1D) we will use polar coordinates in the parameter space A
=
lR2 .fVriting \ '" (A 1 '\2)
=
p(cos8,s-in8), we have
F(a,pcos8,psin8)
pF(a,8,p)
( 11 )
with
F(a,8,p)
1 D\F(a,TP(cos8,sin8)).(cos8,sin8)dT .
fa
( 12)
327
The function F(a,e,p) is of class CZ, Zn/k-periodic in a, Zn-periodicin 6, and ,...;
F(a,e+n,-p) = -F(a,6,p)
V(a,e,p) .
(13)
Using (Z) we find F(a,e,O) = Pl(a)cos6 + Pz(a)sin6
V(a,6) .
(14 )
with zn 3h .. PiCa) = 0 3Ai (t-a,xO(t) ,xO(t) ,0)xO(t)dt
J
1,Z
i
(15 )
For p f 0 equation (10) is equivalent to F(a,e,p) = 0
(16)
Now we make the following hypothesis (Hl)
(Pl(a),PZ(a)) f (0,0)
Va
E
lR .
( 17)
Z Z l/Z Let n(a) = (Pl(a)+PZ(a)) then it is a classical result (see e.g. Knobloch and Kappel [129]) that under the hypothesis CH1) there exists a CZ_ function I'; : lR -+ lR such that : (-PZ(a),Pl(a)) = n(a) (cos I'; (a) ,sinl';(a))
Va
E
lR .
( 18)
Since p.(a) is Zn/k-periodic, we will have 1 , I'; (a
+ Zn/k)
I'; (a)
+ Znm
Va
E
lR ,
( 19)
for some m E ?l. 3F Then F(a,e,O) = n(a)sin(e-I';(a)) , F(a,l';(a),O) o and ae(a, I'; (a) ,0) f O. It follows from the implicit function theorem, the periodicity properties of F and (13), that there exist a Po > 0 and a C2-function e* : lRX]-PO'PO[ -+lR such that 328
(i) ((a,e,p) I F(a,e,p) =0 and Ipi
OJ
t
Sj (mod 'IT), for all i,j
==
1,Z, ... ,r ; i f j .
sgn s"(a j ). It follows from (HZ) and (H3) that we can find 0 such that for all EEl 0, EO [, for all j == 1, Z, ... ,r and for all ==
9.,
E 7L
we have : (Z4 )
Moreover, if e
t
Sj (mod 11) for al1 j
Consider now the equation 330
==
1,Z, ... ,r, then NO(G) is even.
o .
p)
(25)
Because of (l12) and 8* (a,O)
= I;
(a) there exists some Po > 0 such that the
solution set of (25) for I P [ < Po will have the form
(p)+
,PI i Ip[-
m.,
1,2, ... ,r), such that the following holds , with 0
=
< P < PO' equation (3) has a
fini te num..1-Jer of different in-periodic solutions, near yo; we denote this Ilwnber by N(A) (ii) N(p,G) (iii) if 8
f
;?
I;j
2!m[, Yep, (mod TIl, j
:'Ii(p,G) ;
, where mE Z is as in (21) ; =
1,2, ... ,r, then limN(p,8) p-rO
=
N (8) 0
331
(iv) the curves : C. = {(pcose.(p),psine.(p)) J
J
J
i ipi <po}'
j
= 1,2, ... ,r
divide the neighbourhood {A EJR2 i IIAII < PO} of the origin in JR2 into 2r cone-like regions; in the interior of each of these regions N(A) is constant and even ; (v) if A crosses one of the curves C., in the direction of increasing J e, then N(A) increases by two'if OJ = +1, and decreases by two if 0·
J
=
-1 ;
(vi) on Cj \ {OJ N(A) is constant, odd-valued, and equal to NO(sj) ; (vii) the curve Cj is tangent to the line {(PCOS Sj , ps~nsj) i p EJR}; the union of the curves Cj form the bifurcation set for the problem. 0 In [ 90] Hale and Taboas discuss the bifurcation of 2IT-periodic solutions of a special case of (3) near a family YO generated by a solution Xo of (5) for which the condition (4.6) is not satisfied.
332
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Index
autonomous equation 282 150,254 axisymmetric solutions
irreducible representations 71 isotropy subgroup 187,212,309
bifurcation equation 92 bifurcation function 92 bifurcation point 2,153 buckling problem 47,120,220
Kondrachov compactness theorem Krein-Rutman theorem 17
Casimir operator 82 circular plate 52,145,225 compact operator 14 cone 16 conservative system 27,109,115, 125,138,320 cylindrical plate 60,123,146,221 Dirichlet problem
32,117,141,147, 231,251 degenerate Hopf bifurcation 300
elliptic operator 31 equivalent representations 70 equivariant mappings 8,94 equivariant projections 66,96 Floquet exponents, multipliers 240 Fourier expansion 284 frequency response curve 110 functional differential equations 282,291
48
Lax-Milgram theorem 16 Liapunov Center theorem 304 Liapunov-Schmidt method 4,8S,313 Lie group 79,308 local bifurcation 2 maximum principle monodromy matrix
31,32,45 24,239
non-generic bifurcation 179,183 nonresonance condition 241,296,297 O(n) 73 orbit 8,309 ordered Banach space 17 orthogonal representations
70
periodic solutions 22,102,138 positive operator 17 pseudo-inverse 14,26
rank theorem 19 rectangular plate 54,122,185 reduced bifurcation equation 217 ,249,250 Garding's inequality 40 representation 63 generalized solutions 39,49,56 restricted bifurcation problem 4 generic bifurcation 4,153,174 reversible system 105,301 generic conditions 7,158,176,269,279 Riesz representation theorem 16 generic Hopf bifurcation 273 Schauder estimates 34 group action 309 Schur's lemma 71 simple eigenvalue 218 Haar measure 65 Hold"er continuous functions 33 singularity theory 8,158,176 Sobolev spaces 38 Holder inequality 48 solution set Hopf bifurcation 261,281,296 SO(n) 73 spherical harmonics 85,148,253 imbedding theorems 34,38,48 114,237 subharmonic solutions implici t function theorem 18 inverse function theorem 19 349
topological complement 12 topological group 63 transition matrix 23 transversality conditions 7,158,175,242 tubular neighbourhood 309 unitary representations
67
variational equation 27 variation-of-constants formula von K~mntm equations 47,120 weak derivative weak resonance
350
37 251
23
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Titles in this series (continued) Wave propagation in viscoelastic media FMainardi Nonlinear partial differential equations and their applications: College de France Seminar. Volume I H Brezis and J L Lions Geometry of Coxeter groups HHiller Cusps of Gauss mappings T Banchoff, T Gaffney and C McCrory An approach to algebraic K-theory A JBerrick Convex analysis and optimization J-P Aubin and R B Vintner Convex analysis with applications in the differentiation of convex functions JRGiles Weak and variational methods for moving boundary problems C M Elliott and J R Ockendon Nonlinear partial differential equations and their applications: College de France Seminar. Volume II H Brezis and J L Lions Singular systems of differential equations II SLCampbell Rates of convergence in the central limit theorem Peter Hall Solution of differential equations by means of one-parameter groups JMHill Hankel operators on Hilbert space S CPower Schrbdinger-type operators with continuous spectra M S P Eastham and H Kalf Recent applications of generalized inverses S L Campbell Riesz and Fredholm theory in Banach algebra B A Barnes, G J Murphy, M R F Smyth and TTWest
68 69 70
71 72 73 74
75
Evolution equations and their applications F Kappel and W Schappacher Generalized solutions of Hamilton-Jacobi equations PLLions Nonlinear partial differential equations and their applications: College de France Seminar. Volume III H Brezis and J L Lions Spectral theory and wave operators for the Schrbdinger equation AMBerthier Approximation of Hilbert space operators I D A Herrero Vector valued Nevanlinna Theory H JW Ziegler Instability, nonexistence and weighted energy methods in fluid dynamics and related theories B Straughan Local bifurcation and symmetry A Vanderbauwhede