BIFURCATION THEORY I D RPPLICflTIONS
WORLD SCIENTIFIC SERIES ON NONLINEAR SCIENCE Editor: Leon O. Chua University of California, Berkeley Series A.
MONOGRAPHS AND TREATISES
Volume 35:
Introduction to Control of Oscillations and Chaos A. L Fradkov & A. Yu. Pogromsky
Volume 36:
Chaotic Mechanics in Systems with Impacts & Friction B. Blazejczyk-Okolewska, K. Czolczynski, T. Kapitaniak & J. Wojewoda Invariant Sets for Windows — Resonance Structures, Attractors, Fractals and Patterns A. D. Morozov, T. N. Dragunov, S. A. Boykova & O. V. Malysheva
Volume 37:
Volume 38:
Nonlinear Noninteger Order Circuits & Systems — An Introduction P. Arena, R. Caponetto, L Fortuna & D. Porto
Volume 39:
The Chaos Avant-Garde: Memories of the Early Days of Chaos Theory Edited by Ralph Abraham & Yoshisuke Ueda
Volume 40:
Advanced Topics in Nonlinear Control Systems Edited by T. P. Leung & H. S. Qin
Volume 41:
Synchronization in Coupled Chaotic Circuits and Systems C. W. Wu
Volume 42:
Chaotic Synchronization: Applications to Living Systems £ Mosekilde, Y. Maistrenko & D. Postnov
Volume 43:
Universality and Emergent Computation in Cellular Neural Networks R. Dogaru Bifurcations and Chaos in Piecewise-Smooth Dynamical Systems Z T. Zhusubaliyev & E. Mosekilde
Volume 44: Volume 45:
Bifurcation and Chaos in Nonsmooth Mechanical Systems J. AwrejcewiczA C.-H. Lamarque
Volume 46:
Synchronization of Mechanical Systems H. Nijmeijer & A. Rodriguez-Angeles
Volume 47:
Chaos, Bifurcations and Fractals Around Us W. Szemplihska-Stupnicka
Volume 48:
Bio-Inspired Emergent Control of Locomotion Systems M. Frasca, P. Arena & L. Fortuna
Volume 49:
Nonlinear and Parametric Phenomena V. Damgov
Volume 50:
Cellular Neural Networks, Multi-Scroll Chaos and Synchronization M. E. Yalcin, J. A. K. Suykens & J. P. L Vandewalle
Volume 51:
Symmetry and Complexity K. Mainzer
Volume 52:
Applied Nonlinear Time Series Analysis M. Small
Volume 53:
Bifurcation Theory and Applications T. Ma & S. Wang
Series Editor: Leon 0. Chua
BIFURCATION THEORY RND APPLICATIONS Tian Ma Sichuan university, China
Shouhong Wang Indiana University, USA
YJ5 World Scientific N E W JERSEY • L O N D O N • S I N G A P O R E • BEIJING • S H A N G H A I • H O N G K O N G • TAIPEI • C H E N N A I
Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.
BIFURCATION THEORY AND APPLICATIONS Copyright © 2005 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.
ISBN 981-256-287-7 ISBN 981-256-352-0 (pbk)
Printed in Singapore by World Scientific Printers (S) Pte Ltd
Dedicated to our familes Li, Jiao, Ping, Wayne, and Melinda
Preface
This book provides an introduction to a newly developed bifurcation theory and its applications to dynamical systems and partial differential equations (PDEs) from science and engineering. The first two chapters of the book contain a brief introduction to the standard bifurcation theory for nonlinear PDEs. The treatment of the classical theorems is unified by the Lyapunov-Schmidt reduction and the center manifold reduction procedures. The next four chapters introduce a new bifurcation theory developed recently by the authors. This theory is centered at a new notion of bifurcation, called attractor bifurcation for nonlinear evolution equations. The main ingredients of the theory include a) the attractor bifurcation theory, b) steady state bifurcation for a class of nonlinear problems with even order nondegenerate nonlinearities, regardless of the multiplicity of the eigenvalues, and c) new strategies for the Lyapunov-Schmidt reduction and the center manifold reduction procedures. With the bifurcation theory, many long standing bifurcation problems in science and engineering are becoming accessible, and are treated in the last four chapters of the book. In particular, applications are made for variety of PDEs from science and engineering, including, in particular, the Kuramoto-Sivashinshy equation, the Cahn-Hillard equation, the GinzburgLandau equation, Reaction-Diffusion equations in Biology and Chemistry, the Benard convection problem, and the Taylor problem. The applications provide, on the one hand, general recipes for other applications of the theory addressed in this book, and on the other hand, full classifications of the bifurcated attractor and the global attractor as the control parameters cross certain critical values, dictated usually by the eigenvalues of the linearized problems. It is hoped that the book will be helpful in advancing the study vii
viii
Bifurcation Theory and Applications
of nonlinear dynamics for problems in science and engineering. We would like to acknowledge explicitly the great impact on our career of Professors Wenyuan Chen, Jacques-Louis Lions, Louis Nirenberg and Roger Temam. We have greatly benefited from discussions with Hari Bercovici, Jerry Bona, Zhimin Chen, Ciprian Foias, Michael Ghil, Michael Jolly, Benoit Perthame, Roger Temam, Xiaoming Wang, and Kevin Zumbrun, whom we warmly thank. Special thanks are due to three students, Willie Hsia, Jungho Park, and Masoud Yari, at Indiana University, who read earlier drafts of the book and made numerous helpful corrections and suggestions. The first draft was patiently typed by Ginny Jones and Shan Ma. Also we are grateful to Wen Masters and Reza Malek-Madani of Office of Naval Research for their constant support and encouragement. The final touch of the book was done while the second author was visiting DMA (Departement de mathematiques et applications) and Departement TerreAtmosphere-Ocean (TAO) at Ecole Normale Superieure (ENS) Paris; their hospitality and support are acknowledged. Also, we wish to thank our editors, Qin Jing and Rok Ting Tan, at World Scientific for their support and patience. The research presented in this book was supported in part by grants from the Office of Naval Research, and from the National Science Foundation. Tian Ma and Shouhong Wang Bloomington, Indiana April, 2005
Contents
vii
Preface 1.
2.
Introduction to Steady State Bifurcation Theory
1
1.1 Implicit Function Theorem 1.2 Basics of Topological Degree Theory 1.2.1 Brouwer degree 1.2.2 Basic theorems of Brouwer degree 1.2.3 Leray-Schauder degree 1.2.4 Indices of isolated singularities 1.3 Lyapunov-Schmidt Method 1.3.1 Preliminaries 1.3.2 Lyapunov-Schmidt procedure 1.3.3 Normalization 1.4 Krasnosel'ski Bifurcation Theorems 1.4.1 Bifurcation from eigenvalues with odd multiplicity . 1.4.2 Krasnosel'ski theorem for potential operators . . . . 1.5 Rabinowitz Global Bifurcation Theorem 1.6 Notes
1 2 2 4 5 7 8 8 9 12 13 13 14 17 19
Introduction to Dynamic Bifurcation
21
2.1 Motivation 2.2 Semi-groups of Linear Operators 2.2.1 Introduction 2.2.2 Strongly continuous semi-groups 2.2.3 Sectorial operators and analytic semi-groups 2.2.4 Powers of linear operators
21 23 23 25 26 28
ix
x
3.
4.
Bifurcation Theory and Applications
2.3 Dissipative Dynamical Systems 2.4 Center Manifold Theorems 2.4.1 Center and stable manifolds in R™ 2.4.2 Center manifolds for infinite dimensional systems . . 2.4.3 Construction of center manifolds 2.5 Hopf Bifurcation 2.6 Notes
29 32 32 34 37 38 40
Reduction Procedures and Stability
41
3.1 Spectrum Theory of Linear Completely Continuous Fields . 3.1.1 Eigenvalues of linear completely continuous fields . . 3.1.2 Spectral theorems 3.1.3 Asymptotic properties of eigenvalues 3.1.4 Generic properties 3.2 Reduction Methods 3.2.1 Reduction procedures 3.2.2 Morse index of nondegenerate singular points . . . . 3.3 Asymptotic Stability at Critical States 3.3.1 Introduction to the Lyapunov stability 3.3.2 Finite dimensional cases 3.3.3 An alternative principle for stability 3.3.4 Dimension reduction 3.4 Notes
41 41 44 50 53 56 56 62 66 66 67 70 72 74
Steady State Bifurcations
75
4.1 Bifurcations from Higher-Order Nondegenerate Singularities 75 4.1.1 Even-order nondegenerate singularities 75 4.1.2 Bifurcation at geometric simple eigenvalues: r = 1 . 83 4.1.3 Bifurcation with r = k = 2 85 4.1.4 Reduction to potential operators 90 4.2 Alternative Method 92 4.2.1 Introduction 92 4.2.2 Alternative bifurcation theorems 94 4.2.3 General principle 98 4.3 Bifurcation from Homogeneous Terms 100 4.4 Notes 103 5.
Dynamic Bifurcation Theory: Finite Dimensional Case
105
xi
Contents
6.
7.
5.1 Introduction 5.1.1 Pendulum in a symmetric magnetic field 5.1.2 Business cycles for Kaldor's model 5.1.3 Basic principle of attractor bifurcation 5.2 Attractor Bifurcation 5.2.1 Main theorems 5.2.2 Stability of attractors 5.2.3 Proof of Theorems 5.2 and 5.3 5.2.4 Structure of bifurcated attractors 5.2.5 Generalized Hopf bifurcation 5.3 Invariant Closed Manifolds 5.3.1 Hyperbolic invariant manifolds 5.3.2 S1 attractor bifurcation 5.4 Stability of Dynamic Bifurcation 5.5 Notes
105 105 110 112 114 114 116 119 123 127 129 129 132 138 149
Dynamic Bifurcation Theory: Infinite Dimensional Case
151
6.1 Attractor Bifurcation 6.1.1 Equations with first-order in time 6.1.2 Equations with second-order in time 6.2 Bifurcation from Simple Eigenvalues 6.2.1 Structure of dynamic bifurcation 6.2.2 Saddle-node bifurcation 6.3 Bifurcation from Eigenvalues with Multiplicity Two 6.3.1 An index formula 6.3.2 Main theorems 6.3.3 Proof of main theorems 6.3.4 Case where k > 3 6.3.5 Bifurcation to periodic solutions 6.4 Stability for Perturbed Systems 6.4.1 General case 6.4.2 Perturbation at simple eigenvalues 6.5 Notes
152 152 154 160 160 163 165 165 169 172 184 184 188 188 191 194
....
Bifurcations for Nonlinear Elliptic Equations
197
7.1 Preliminaries 7.1.1 Sobolev spaces 7.1.2 Regularity estimates
197 197 200
xii
Bifurcation Theory and Applications
7.1.3 Maximum principle 201 7.2 Bifurcation of Semilinear Elliptic Equations 202 7.2.1 Transcritical bifurcations 202 7.2.2 Saddle-node bifurcation 207 7.3 Bifurcation from Homogenous Terms 209 7.3.1 Superlinear case 209 7.3.2 Sublinear case 210 7.4 Bifurcation of Positive Solutions of Second Order Elliptic Equations 213 7.4.1 Bifurcation in exponent parameter 214 7.4.2 Local bifurcation 222 7.4.3 Global bifurcation from the sublinear terms 231 7.4.4 Global bifurcation from the linear terms 236 7.5 Notes 240 8. Reaction-Diffusion Equations
241
8.1 Introduction 241 8.1.1 Equations and their mathematical setting 241 8.1.2 Examples from Physics, Chemistry and Biology . . . 243 8.2 Bifurcation of Reaction-Diffusion Systems 246 8.2.1 Periodic solutions 246 8.2.2 Attractor bifurcation 248 251 8.3 Singularity Sphere in 5m-Attractors 8.3.1 Dirichlet boundary condition 251 8.3.2 Periodic boundary condition 256 8.3.3 Invariant homological spheres 258 8.4 Belousov-Zhabotinsky Reaction Equations 259 8.4.1 Set-up 259 8.4.2 Bifurcated attractor 260 8.5 Notes 265 9. Pattern Formation and Wave Equations
267
9.1 Kuramoto-Sivashinsky Equation 9.1.1 Set-up 9.1.2 Symmetric case 9.1.3 General case 9.1.4 S1-invariant sets 9.2 Cahn-Hillard Equation
267 267 268 271 273 275
Contents
9.3
9.4
9.5 9.6
9.2.1 Set-up 9.2.2 Neumann boundary condition 9.2.3 Periodic boundary condition 9.2.4 Saddle-node bifurcation Complex Ginzburg-Landau Equation 9.3.1 Set-up 9.3.2 Dirichlet boundary condition 9.3.3 Periodic boundary condition Ginzburg-Landau Equations of Superconductivity 9.4.1 The model 9.4.2 Attractor bifurcation 9.4.3 Physical remarks Wave Equations 9.5.1 Wave equations with damping 9.5.2 System of wave equations Notes
10. Fluid Dynamics
xiii
275 276 286 290 291 291 293 296 297 297 302 315 322 322 324 325 327
10.1 Geometric Theory for 2-D Incompressible Flows 327 10.1.1 Introduction and preliminaries 327 10.1.2 Structural stability theorems 327 10.2 Rayleigh-Benard Convection 330 10.2.1 Benard problem 330 10.2.2 Boussinesq equations 331 10.2.3 Attractor bifurcation of the Rayleigh-Benard problem 335 10.2.4 2-D Rayleigh-Benard convection 341 10.3 Taylor Problem 343 10.3.1 Taylor's experiments and Taylor vortices 343 10.3.2 Governing equations 343 10.3.3 Stability of secondary flows 349 10.3.4 Taylor vortices 354 10.4 Notes 365 Bibliography
367
Index
373
Chapter 1
Introduction to Steady State Bifurcation Theory In this chapter, we present some classical theorems on steady state bifurcations, including the Lyapunov-Schmidt procedure, bifurcation theorems from eigenvalues of odd multiplicity, and bifurcation theorems for potential operators. The presentation we adopt here is to use the Lyapunov-Schmidt procedure to link all these theorems in a natural way. The version of the Lyapunov-Schmidt procedure presented here differs slightly from the one given in classical textbooks. The later is done by decomposing the space into the direct sum of the eigenspace and its complement. While the Lyapunov-Schmidt procedure presented here is based on the decomposition of the space into the direct sum of the generalized eigenspace and its complement. This Lyapunov-Schmidt procedure is more nature, and much more convenient to study steady state bifurcations. In fact, it is this difference, together with other ingredients, including in particular the spectral theorem in Chapter 3, that made many problems more accessible. Another important ingredient for the Lyapunov-Schmidt procedure presented here is the introduction of the normalization of the bifurcation equation (1.19), which is crucial for the new steady state bifurcation theory from higher order terms regardless of the multiplicity of the linearized eigenproblems given in Chapter 4. 1.1
Implicit Function Theorem
Let X, Y, Z be Banach spaces, U C X x Y an open set, and F : U —> Z a mapping. Let (xo,yo) € U satisfy F(xo,yo)=O. 1
(1.1)
2
Bifurcation Theory and Applications
We consider the solvability near (xo,yo) of the equation F(x,y) = 0.
(1.2)
The following well known implicit function theorem is of fundamental importance in the bifurcation theory.
Theorem 1.1
(Implicit Function Theorem). Let F e Ck(U, Z) be k-
th order differentiable (k > 1), and satisfy (1.1). Assume that the derivative operator DxF(xo,yo) : X —> Z has a bounded inverse. Then, the following assertions hold true. (1) There exist a neigborhood V C X of XQ and a neighborhood W CY of 2/o such that as y S W, (1.2) has a unique solution x = $(y) 6 V. (2) The mapping $ : W —> V is k-th differentiable, especially if F is analytic, then $ is also analytic. (3) If DyF(xo,yo) — 0, then we have $/(2/o) = 0,
i.e.$(y)=o(\\y-yo\\).
(1.3)
Remark 1.1 Actually, if the derivations of F with respect to y vanish up to m-th order at (xo,yo)'-
F$n)(xo,yo) = O,
Vl Y a linear homeomorphism, and B\ : X —> Y a linear compact operator, then there exist u £ X with u ^ 0 such that L\ou = 0. 1.3.2
Lyapunov-Schmidt
procedure
We begin with an example to introduce the Lyapunov-Schmidt procedure. Consider a system of two algebraic equations given by ( anx\ + ai2x2 + fi(xi, x2) = 0, [a 2 lZl +0,22X2 + h{xi,X2) = 0 ,
(1.8)
where a^ = a*,- (A) are continuous functions of A, and fi, f2 are C°° functions satisfying that
(h(x)J2(x))=o(\x\).
10
Bifurcation Theory and Applications
Obviously, (x, A) = (0, A) is a trivial solution of (1.8). We shall use the Lyapunov-Schmidt procedure to investigate the bifurcation of (1.8). For simplicity, we assume that the matrix A
=
fan(X) a 12 (A)\ Wi(A) a22(X)J
has two eigenvalues in a neighborhood of Ao as follows
01 = A - Ao,
fa
= 1.
Under a coordinate transformation, the equations (1.8) are rewritten as (X-Xo)x1+Fi{xux2)
= 0,
(1.9) (1.10)
x2+F2(x1,x2)=0,
where Fi(x) = o(\x\), i = 1,2. By the implicit function theorem-Theorem 1.1, (1.10) has a solution near x\ = 0: X2=g{xx),
g(Xl) = o(\Xl\).
(1.11)
Inserting (1.11) into (1.9) we arrive at
U\-\0)x1+F1(x1,g(x1))
= 0,
\F1(x1,g(x1))=o(\x1\).
(1.12)
Thus, the bifurcation of (1.8) is equivalent to that of (1.12). By the index theorem-Theorem 1.7, we find f 1 ind((A-A o )id + J Fi,0)=ind((A-A o )id,0) = ^ [ -1
if A > Ao, ' if A < Ao,
which implies that (1.12) has a bifurcation from (0, Ao). Hence, (0, Ao) is a bifurcation point of (1.8). We now introduce the Lyapunov-Schmidt method in general setting. Consider u - XAu + G(u, A) = 0,
(1.13)
where u S X, X a Banach space, A : X —> X a linear compact operator, and G : I x R - t I a continuous mapping satisfying (1.6).
Introduction to Steady State Bifurcation Theory
11
Assume that A^ 1 is a real eigenvalue of A with algebraic multiplicity m > 1. Thus, the space X can be decomposed into the direct sum of two invariant subspaces of A as follows
X = E0®E1, Eo = ( J {x G X | (id - \0A)nx = 0}, n&N
dimi?o = m. The linear operator A : X —» X can be decomposed into A = A0 + Ai, Ao = A\Eo : Eo -* Eo, Al=A\El
: Ei-ȣ?i.
Let Po : A" —> Eo and Pi = id — PQ : X —* Ei be the canonical projections. Then (1.13) is decomposed into x-\Aox
+ PoG(x + y,X)=O,
j/ - \AlV + PiG(x + y, X) = 0,
x e Eo,
(1.14)
ye EL
(1.15)
It is clear that id — XQAI : E\ —» E\ is invertible. Based on the implicit function theorem, (1.15) can be solved near x — 0 and y — 0: 2/= 1. Then the bifurcation problem of (1.13) near X = Ao is equivalent to that of (1.17). We remark here that the Lyapunov-Schmidt procedure presented here differs slightly from the one usually presented in classical textbooks. The later is done by decomposing X into the direct sum of the kernel XQ of id — Ao A and its complement X\: X = XO®XU
Xo = ker(id - X0A).
12
Bifurcation Theory and Applications
While the Lyapunov-Schmidt procedure presented here is based on the decomposition X = Eo © E\ with Eo being generalized eigenspace. As we shall, it is easy to see that this Lyapunov-Schmidt procedure is more nature, and much more convenient to study bifurcation of (1.13). In fact, it is this difference, together with other ingredients presented in this book that made many problems more accessible. 1.3.3 Normalization Let the nonlinear term G : X x R1 —> X be C°°. Then there is a k > 2 such that
G(u,\) = JTGn(u,\), n=k
where G n (u,A) is an n-multilinear mapping, and Gn(au,\)=anGn(u,\),
VaeR 1 .
By (1.16), y(x, A) is a higher order function of x. Hence, (1.17) has the following form x-\Aox
+ PoGh(x,X)+o(\\x\\k)=Q,
k>2.
(1.18)
If x = 0 is an isolated singular point of PoGk, then the bifurcation problem of (1.18) is reduced to that of the following equation: x-\Aox
+ PoGk(x,X)=0.
(1.19)
There are many bifurcation problems in science and engineering, where the bifurcation equation (1.19) can be explicitly calculated, providing an important step toward to their bifurcation analysis. For example, we consider a special case where A : X —> X is a symmetric linear compact operator, X is a Hilbert space, A^1 is an eigenvalue of A with multiplicity two, and e 1; e2 are the eigenvectors corresponding to Ao, which are orthogonal
Let k = 2 and G^ be the second order multilinear operator of G. The equation (1.19) can be expressed as the following system of 2-dimensional
13
Introduction to Steady State Bifurcation Theory
algebraic equations j (1 - XXQ V I + anxl + 1a\2xix2 + a\2x\ = 0, { (1 - AA^x)x2 + a\\x\ + 1a\2x±X2 + a\2x\ = 0, where 4 = 4W
= \ [(G2(ei,ej,X),ek) + (G2(ej,ei,X),ek)}.
Remark 1.3 The method that reduces the bifurcation equation (1.17) to its normalized form (1.19), together with the spectrum theorem and the attractor bifurcation theorems developed in [Ma and Wang, 2004e; Ma and Wang, 2004d], is a very useful tool in the bifurcation analysis in many problems. The bifurcation theory with applications developed by the authors is summarized in this book and as well as in a forthcoming book on hydrodynamic stability and bifurcation. 1.4 1.4.1
Krasnosel'ski Bifurcation Theorems Bifurcation from eigenvalues with odd multiplicity
When the eigenvalue AQ""1 of A has odd multiplicity, the following wellknown Krasnosel'ski bifurcation theorem says that the equation (1.13) has a bifurcation from (0, Ao). Theorem 1.10
(Krasnosel'ski Theorem). Under the condition (1.6),
if A : X —> X is a linear compact operator, and XQ1 6 M.1 is a real eigenvalue of A with odd algebraic multiplicity, then (u,X) = (0, Ao) is a bifurcation point of (1.13).
Proof. This theorem can immediately be derived by the index theorem (Theorem 1.7) and the Lyapunov-Schmidt theorem (Theorem 1.9). Indeed, without loss of generality, we assume that Ao > 0. Then the bifurcation equation (1.17) reads f x - XAox + g{x, X) = 0, \g(x,X)
= P0G(x + y(x,X),X)
= o(\\x\\).
{
By Theorem 1.7, we have ind(id - XA0 + g, 0) = ind(id - XA0,0)=\1
[ - 1
if A < ^o, if A > A o .
'
'
14
Bifurcation Theory and Applications
It implies that (a;, A) = (0,A0) is a bifurcation point of (1.20). Thus, by Theorem 1.9 we obtain the Krasnosel'ski theorem. D Consider a special case where the algebraic multiplicity m = 1 of A^"1, i.e. A^1 is a simple eigenvalue of A. The following theorem is due to Crandall and Rabinowitz. Theorem 1.11 Assume that the conditions of Theorem 1.10 hold true. If \Q1 is a simple eigenvalue of A, then the equation (1.13) bifurcates from (0, Ao) to exactly two branches T\ and T2, which are of the following form T:(ux,X),
ux=te
+ tv(t),
A = Ao + /*(*),
where v(t) G X, fi(t) G R1 are continuous on t with v(0) = 0, n(0) = 0, and e is the eigenvector of A corresponding to XQ . This theorem can be derived from (1.20). In fact, if g = 0, then for any t >0, Ti : ux =te + y(te,X0), r 2 : ux — -te + y(-te, Ao), where y(x,X) is given by (1.16). If g ^ 0 in (1.20), and g(re, A) = p(r, A)e, p(r, A) has the Taylor expansion near r = 0 as follows p(r, A) = a(X)rk
+ o(\T\k),
k>2,
a{X0) ± 0,
then the equation (1.20) can be written as (1 - A A ^ ) T + Q(A)T* + O(|T|*) = 0,
which has two branches of solutions as shown in Figure 1.2. 1.4.2
Krasnosel'ski theorem for potential
operators
Let X be a Hilbert space. For the equation (1.13) we assume that (Ai) A : X —> X is a self-adjoint linear compact operator, and (A2) there is a functional F G Cr(X x M 1 ^ 1 ), with DuF(u, A) = G{u, A),
and G(u, A) satisfies (1.6).
{u, A) G X x R 1 ,
15
Introduction to Steady State Bifurcation Theory
^-
^. AQ
N.
A,
J XQ r
r2
2
(a)
(b) T
*"
^"^
r
2
(c) Fig. 1.2
(a) Case: fc =odd, a(Ao) > 0; (b) case: k =odd, Q(AO) < 0; (c) case: k =even.
The following theorem is due to Krasnosel'ski. Theorem 1.12 Under the hypotheses (A\) and (A2), if A^"1 e R1 is an eigenvalue of A, then the equation (1.13) has at least two bifurcated branches from (u, A) = (0, Ao). To see this theorem, we assume that the eigenvalue Ao has multiplicity m > 1 with orthogonal eigenvectors {ei, • • • , e m } , F = J2™=k+i Fn, k>2, and Fn is the n-multilinear functional of F. Then, the bifurcation equation
16
Bifurcation Theory and Applications
of (1.13) is of the form / m
£)
(1 - XX^Xi + ~ Fk+1 where x = ( x i , • • • , xm)
5>,e,,A
\
+ o(|x|fc)=0,
(1.21)
E M™.
If x = 0 is an isolated singular point oiVFk+i(J2]Li xj^j, -V)), it suffices to consider the bifurcation of the following system of equations
(l-XXo1)x + VFk+1 lj2xJeJ'X)
C 1 - 22 )
=°-
It is clear that the critical points (z(A), A) of the functional
1(1 - XX^)\x\2 + Fk+1 I jpxjej,\\
(1.23)
are the bifurcated solutions of (1.22). One can obtain the critical points of (1.23) by the following method. Let (ai, • • • , am) G Km with |a| 2 = 1 satisfy / m
\
Fk+i ^ Q i e i , A \i=i
/ m
= max Fk+i /
P u t t i n g ( x i , - - - , x m ) = {ait,---
|x|
- 1
,amt)
±(1 - XX^t2 + Fk+i
\
^^e^A \fc=i
. /
in (1.23) we get
(JTaiei,x\tk+1.
Without loss of generality, let Fk+i(Y^Lx a^i, Ao) > 0. Then, for A > Ao we take
tkx~l = (AAo1 - l)/(fe + l)F fc+1 f f ; aiei, A j .
(1.24)
Thus, this point x(X) = (ait\, • • • , am*A) is a critical point of (1.23). In the same fashion, we can obtain another critical point of (1.23) by taking a £ R m as follows / m
Fk+i
Vaiei,A
\
/ m
= min Fk+1 ^2xie^X
\
•
17
Introduction to Steady State Bifurcation Theory
When the operator G = DF is odd, i.e. G(u, A) = - G ( - « , A),
V u G X, A E R1,
we have the following theorem, due to D.C. Clark. Theorem 1.13 Under the hypotheses (A\) and (A2), if G(u,X) is an odd operator, and XQ1 is an eigenvalue of A with multiplicity m, then there are at least m distinct pairs of bifurcated branches of (1.13) from (u, A) = (0,A0). The basic idea offindingcritical points x(A) of (1.23) is as follows. We take (ai, • • • , am) £ Rm such that F
k+\
/ m
Y]onei,\
\~T^
\ /
/ m
= minm
max Fk+\ I Y]xiei,X
K'-cK ieK'-, |a:|=i
\
^
\ I
.
(1-25)
Because Fk+i is even, the values (ai,--- ,am) can be derived for each 1 < r < m, and the m distinct pairs of points xr(X)
— (±arit\,
••• ,
±armt\)
are the critical points of (1.23), where ar = (ari, • • • ,arm) satisfies (1.25) and t\ satisfies (1.24). 1.5
Rabinowitz Global Bifurcation Theorem
In this section, we introduce the Rabinowitz global bifurcation theorem. Let B = {(u, A) G X x R1 I u ^ 0, (u, A) satisfy (1.13)}. Theorem 1.14 Let G : X x R 1 —> X be a compact operator satisfying (1.6), and A : X —> X a linear compact operator. IfX^ is a real eigenvalue of A with odd algebraic multiplicity, then the connected component E c 5 which contains (O,Ao) satisfies one of the following assertions: (1) E is unbounded in X x R , or (2) E contains odd number of points (0, A*) ^ (0, Ao) such that A" 1 are the eigenvalues of A with odd algebraic multiplicities.
This theorem is, in essence, a corollary of the homotopy invariance of the Leray-Schauder degree. The assertions (1) and (2) of Theorem 1.14 amount to saying that the connected component E C B containing (0, Ao) must be
18
Bifurcation Theory and Applications
one of the two types as shown in Figure 1.3(a) and (b). If otherwise, then S must be bounded in X x R1 as shown in Figure 1.4. By the basic properties of the Leray-Schauder degree, we immedaitely deduce a contradiction. More precisely, by the excision property (see Figure 1.4), we have deg(id - X2A + G, BR, 0) = deg(id - X2A + G,Br, 0) (1.26)
= ind(id-X2A,0) = (-lf. Then the homotopy invariance of the degree shows that deg(id - X2A + G, BR, 0) = deg(id -
+ G, BR, 0),
(1.27)
deg(id - X2A + G, Br, 0) = deg(id - XXA + G, Br, 0).
(1.28)
PlA
It follows then from (1.26)-(1.28) that deg(id -
PlA
+ G, BR, 0) = deg(id - XXA + G, BR, 0) = (-1)".
(1.29)
Furthermore, by the index theorem (Theorem 1.7), deg(id - PlA + G, BR, 0) = ind(id - PlA, 0) = (-I)""" 1 , (1.30) where m = odd is the algebraic multiplicity of Ao, and BR = {x £ X | ||a:|| < R}. Thus, we arrive at a contradiction from (1.29) and (1.30). X
iX
•
L
1 :
(a)
-^
(b) Fig. 1.3 (a) £ is unbounded; (b) 2 is bounded.
1*-
Introduction to Steady State Bifurcation Theory
19
X
B
Pi
R
^__J^
/
I
*o
*i|
Br h
Fig. 1.4 The larger rectangular box is BR X [pi,A2], and the smaller rectangular box is Br x [\\, A2].
1.6
Notes
1.1 The proof of the classical implicit function theorem can be found in many standard textbooks; see, among others, [Chen, 1981; Chow and Hale, 1982; Nirenberg, 2001; Zhong et al., 1998]. 1.2 Both the Brouwer degree theory and the Leray-Schauder degree theory can be found in classical books, including, among many others, [Chen, 1981; Krasnosel'skii, 1956; Nirenberg, 2001; Zeidler, 1986]. 1.3 The standard Lyapunov-Schmidt procedures are discussed in [Nirenberg, 2001; Chow and Hale, 1982; Golubitsky and Schaeffer, 1985; Golubitsky et al., 1988]. 1.4 Theorems 1.10 and 1.12, are due to the pioneering work by [Krasnosel'skii, 1956]. Theorem 1.11 is due to [Crandall and Rabinowitz, 1971]. Theorem 1.13 is proved by [Clark, 1975], and is generalized by [Rabinowitz, 1977]. 1.5 The Rabinowitz global bifurcation theorem, Theorem 1.14, is due to [Rabinowitz, 1971].
Chapter 2
Introduction to Dynamic Bifurcation
In this chapter, we briefly present basic theories on semi-groups of both linear and nonlinear operators on Banach spaces, on dissipative dynamical systems, and on center manifold theorems. In addition, the classical Hopf bifurcation for a finite dimensional dynamical system is given based on the center manifold reduction and the classical Poincare-Bendixson theorem. 2.1
Motivation
We start with a simple example given by
• ^ = g + U2-A/(x) V* e( 0,l)cM\ at a; = 0 , l ,
' u= 0
(2-1)
Vxe(0,l),
_u(x,0) = 1 ) i
at a; = 0,1.
(
2-1
)
We know that there exists a number 0 < A < oo such that for all |A| < A, (2.2) has a solution v\(x), called a steady state of (2.1), which depends continuously on A. The stability theory tells us that given a general function / G £ 2 (0,1), there is a Ao (0 < Ao < A), and a neighborhood U\ C L2(0,1) of the steady state v\ for any |A| < Ao, such that for any initial value
oo
=0.
However, when Ao < A the stability will be lost. Then it is nature to study two questions: (1) from what values of A, the steady state v\ will lose its stability, and (2) if Ao is the point from which v\ loses its stability, then what behaviors of the solutions u(x, t) of (2.1) near v\ may occur for t —* oo. The problem (1) is related to the steady state bifurcations introduced in Chapter 1, and both problems (1) and (2) are central topics of dynamic bifurcation. We end this introduction with two remarks. First, let u = v + v\ in (2.1) and (2.2). Then the stability and bifurcation problems for (2.1) and (2.2) near v\ are equivalent respectively to the following problems:
1
v=0 v{x,0) = ip(x)
( 2i3 )
at a; = 0,1, Vz€(0,l),
r g + 2 U A V + u 2 = o v, e( o,D,
{2A)
at a; = 0 , 1 ,
[v = 0
near v = 0, where ip —
H by Av = - ^ ,
Bxv = 2vxv,
Gv = v2,
v e Hi.
(2.5)
Introduction to Dynamic Bifurcation
23
The equations (2.3) and (2.4) are then reformulated into the following operator forms respectively -=L,u
+ Gu,
u€Hlt
( 2 6 )
«(0) = V, and (2.7)
L\u + Gu = 0.
We know that A : Hi —» H is a linear homeomorphism, B\ : Hi —> H a linear compact operator, and G : H\ —> i7 a compact operator. Moreover, we see that -BA : ffi/a -^ H a bounded linear operator, G : Hi/2 —> H a bounded continuous operator, G(u) =
o(\\u\\1/2).
We notice that the equation (2.7) can be rewritten as u-A-1Bu-A~1Gu = 0,
u£H,
which has the same form as (1.13). In fact, many differential equations in Physics, Chemistry and Biology can be put into the abstract form given in (2.6) and (2.7). 2.2 2.2.1
Semi-groups of Linear Operators Introduction
As an example, we continue to consider the equation (2.3). By the SturmLiouville theorem, for each |A| < A, there exist a sequence of eigenvalues {Afc} with Afc —> +oo as k —* oo and eigenvectors {cf>k} C Hi such that
{
d24> — k +2vxk = -\k<j>k,
4>k(0) = 0*(1) = 0, and {4>k} C H constitutes an orthogonal basis of H.
{2g)
24
Bifurcation Theory and Applications
Let u = Y^kLi xk4>k- The equation (2.3) can be written as / I T = -**** + **(«).
*= i,2,..,
(29)
l*fc(0) = Vfc, where V* = fo i> • fadx, Gk(u) = / ^ u2kdx. We know that the solution of (2.9) can be expressed by Jo which implies that the solution of (2.3) is given by oo
u{x,t) = Y,e~Xkt^k+
-t oo e Xk{t T)G u
Y, ~
•/o
fc=l
~
^ )^dT.
(2.10)
fc=l
We find from (2.10) a family of linear bounded operators T(t) : H —» H defined by CXI
OO
A
T(tty = 5>- *Vfe&,
V t > 0 , V = X)V-fc^*.
(2-11)
fc=i fe=i
which enjoy the following semi-group properties: ' T(0) = id : £T -^ F ,
H there exists a one parameter family of linear bounded operators T(t) : H —> H satisfying (2.12), then the solution of the nonlinear equation (d^=Lu
+
G(u),
( 2 i 4 )
I «(0) = V, can be expressed by (2.13). The family of linear bounded operators T(t) is called a semi-group of linear operators.
25
Introduction to Dynamic Bifurcation
2.2.2
Strongly continuous
semi-groups
Definition 2.1 Let X be a Banach space, T(t) : X -> X (0 < t < oo) a one parameter family of bounded linear operators. T(t) is called a strongly continuous semi-group of linear operators, if (1) T(0) = id : X -> X the identity, (2) T{t + s) = T(t)T(s) for every *, s > 0, and (3) lim(_o T(*)a: = a; for each x eX. Let Xi be a Banach space, X\ C X a dense inclusion, and L : Xi —> X a bounded linear operator. We say that L : Xi —> X generates a strongly continuous semi-group T(t) : X —» X, if Lx = ]imT{t)x~X t—o i
= j-T(t)* , dt t=o
VxeXj.
In this case, the operator L : Xi —> X is also called a generator of T(i). The following is the Hille-Yosida theorem. Theorem 2.1 A linear bounded operator L : Xi —+ X generates a strongly continuous semi-group T(t), t > 0 if and only if (1) L is a closed operator, i.e. if xn —» £0, Lxn —> 2/0 wi X , £/ten x n —> zo in X\ and LXQ = 3/0 • ^ T/iere are Ao > 0 and C > 1 suc/i t/iat (Ao, +00) C p(L), and ||(Aid - L ) - " | | < C(A - Ao)-",
V A > Ao, n = 1,2, • • • ,
where p(L) is the resolvent set of L defined by p(L) = {A € C I (Aid - L)-1 : X -» X bounded) .
(2.15)
Remark 2.1 If L : Xi —> X is a linear completely continuous field, then L satisfies Condition (1) in Theorem 2.1. Condition (2) implies that there is a real number Ao > 0, such that for all eigenvalues A € C of L, ReX < Ao. A strongly continuous semi-group T(t) : X —» X generated by L : Xi —» X (Xi C X densely) has the following basic properties. Theorem 2.2 Let T(t) : X —» X (t > 0) be a strongly continuous semigroup generated by L : Xi —> X . Then
26
Bifurcation Theory and Applications
(1) For any UQ £ X\, the following initial value problem ( du
[ u(0) = u0, has a unique solution u(t) = T(t)u0 € Cl([Q, oo), X) n C°([0, oo), Xx). (2) If the equation (2.13) has a solution u(t) £ X\, then the solution u(t) satisfies (2.14). (3) For any xeX, f*T(s)xds € Xu and T(s)xds) = T(t)x - x.
L([
(4) For xeXi,_ T(t)x 6 Xi and T{t)x - T(s)x = f T(r)LxdT = f LT{r)xdT. Js Js 2.2.3
Sectorial operators and analytic semi-groups
The strongly continuous semi-group T(t) : X —> X has a limitation that in general T{t)x $. Xi for t > 0 and x S X. However, for the semigroup of linear operators given by (2.11) the situation is different. Let Xx = H2{0,1) n ^ ( 0 , 1 ) , and X = L2(0,1). Then, t -» T(t)x is analytic for 0 < t and x = 23^=1 xk4>k £ X, and oo
T(t)x
= Y^e~Xktxk X defined b y (2.5), where L = —C is a generator of t h e semi-group T(t) in (2.16), we can define fractional power operators Ca with domains Xa = D(Ca) (a G R ) as follows oo oo 1
{
fc=l r oo
< H^ii _
fc=l
\^\l»xl Lfc=l
oo
cax = YlKxkk £X, V i e i o , fc=i
J
-I i/2
(2.17)
Introduction to Dynamic Bifurcation
27
provided that the eigenvalues of (2.8) satisfy 0 < A^, for any k = 1,2, It is clear that Xa C Xp (a > (3) is a dense and compact inclusion, the norms || • || and || • ||o are equivalent to those of H2(0,1) and L2(0,1) respectively. Furthermore, we have T(t):X^Xa
Vt>0, aeR1,
T(t)Cax = CaT(t)x
Vi£l o ,
nfi _ r/3 _ pot _ rjx+P
£a
Another important property is \\CaT(t)\\0,
which follows from the following computation: ||£ a T(*)||= sup \\£aT(t)x\\ lkllo=i
= sup
r oo
-11/2
Tx^e-^'xll
ll*llo=l U = 1 < max (A£e
1) on (x, y) € Rm x M"- m , and Gi(ar,y, A) =0(1*1,15,1)
(i = 1,2),
V A e R1.
(2.23)
The following is the center manifold theorem. Theorem 2.8 Suppose that all eigenvalues of A have nonnegative (resp. non-positive) real parts, and all eigenvalues of B have negative (resp. positive) real parts. Then for the system (2.22) with the condition (2.23), there exists a Cr function, called the center manifold function, h(-, A) : D —> Rn-m,
D C Rm a neighborhood ofx = 0,
such that h(x, A) is continuous on A, and (1) h{Q1\)=Q,h'x{Q,\)=O; (2) the set Mx = {(x,y) \xeDcRm,
y = h(x,X)},
called the center manifold, is a local invariant manifold of (2.22); (3) if M\ is positively invariant (resp. negatively invariant), namely z(t,tp) € M\ (resp. z(—t,ip) G MA) V t > 0, then M\ is an attracting set of (2.22) (resp. a repelling set), i.e. there is a neighborhood U c M " of M\, as ip € U we have lim dist(z(t, ip), M\) = 0 t—>oo
(resp. lim dist(z(-t,i/>), M\) = 0), t—>oo
where z(t, ip) = (x(t, ip),y(t, ip)) is the solution of (2.22) with the initial value z(O,ip) = ip.
Introduction to Dynamic Bifurcation
33
Property (1) means that the center manifold M\ C W1 is tangent to the eigenspace M.m of A at z = (x, y) = 0. Although, as we know, the local center manifold M\ may not be unique, the following theorem makes it applicable. Theorem 2.9 There is a neighborhood C / c R n of z = 0 such that every invariant set of (2.22) in U belongs to the intersection of all local center manifolds in U. In the following, we introduce the stable manifold theorem. Theorem 2.10 Let all eigenvalues of A have positive real parts, and all eigenvalues of B have negative real parts. Then there exist two unique manifolds Mu and Ms, called the unstable manifold and stable manifold of (2.22) at z = 0, which are characterized by
Mu = lz e Rn I lim S(-t)z = o\ , Ms = lz G W1 I lim S(t)z = OJ , I
t—>oo
J
where S(t) is the semi-group generated by (2.22). Moreover, Mu and Ms are tangent to the eigenspaces of A and B respectively at z = 0: TZ=OMU = R m ,
TZ=OMS = K"- m .
Therefore, the stable manifold Ms and the unstable manifold Mu are transversal at z = 0. Consider the following general system ' dx — < ^=By
=Ax-\-G1{x,y,z,\), + G2(x,y,z,X),
dz — =Cz +
(2.24)
G3(x,y,z,\),
where x G E m , y £ Rk, z € R n , Gi(x,y,z,\) = o(\x\, \y\, \z\) (t = 1,2,3). We have the following invariant manifold theorem. Theorem 2.11 Let the eigenvalues of A have positive real parts, the eigenvalues of B have negative real parts, and the eigenvalues of C have zero real parts. Then, the system (2.24) has three locally invariant manifolds Mu, Ms and Mc, which are tangent to the eigenspaces of A, B, C
34
Bifurcation Theory and Applications
respectively:
T'
A/Tu
u>=O-»w
lUm
T1
— IK ,
H/f3
TD>^
= K ,
1W=QM
T1
A/fC
TO7"1
=K
J.w=oM
,
where Mu is the unstable manifold, Ms is the stable manifold, Mc is the center manifold. Moreover, if Gi (i = 1,2,3) are Cr (r > 1) then Mu, M", Mc are Cr. 2.4.2
Center manifolds for infinite dimensional
systems
Now, we introduce the infinite dimensional version of center manifolds theorems. Let H and Hi be two Hilbert spaces, and Hi C H be a dense inclusion embedding. We consider the nonlinear evolution equation given by tiG^AeR1,
f -^ = Lxu + G(u,A),
1 «(0) = u0,
(2.25)
where L\ : Hi —> H are parameterized linear completely continuous fields depending continuously on A £ R1, which are defined by < A : Hi —> H \ B\ : Hi —> H
linear homeomorphism, parameterized linear compact operators.
(2.26)
We first consider the case in which L\ : H\ —> H is a sectorial operator. Then, one can define fractional power operators L" for a £ l ' with domain Ha = D{L$). Furthermore, we assume that the nonlinear terms G(-,A) : He —> H for some 0 < 8 < 1 are a family of parameterized Cr (r > 1) bounded operators continuously depending on the parameter A g l 1 , such that G{u, A) = o(Htf,),
(0 < 6 < 1), V A G R1.
(2.27)
We assume that the spaces Hi and H can be decomposed into •Hi=E$®E$, tiixnEi < oo, < H = E$@E2, E2 = closure of E% in H,
(2.28)
Introduction to Dynamic Bifurcation
35
for A near Ao, where E^, E2 are the invariant subspaces of L\, i.e. L\ can be decomposed into L\ = C\ ® C2 such that for any A near Ao,
( L2 = L\\E\ . t,2 —> b2 ,
(2.29)
where the eigenvalues of C2 possess negative real parts, and the eigenvalues of L\ possess nonnegative real parts at A = Ao. Thus, for A near Ao, equation (2.25) can be written as
{%=C*1x + Gl(x,y,\), 0, a neighborhood O\ C E$ of x = 0, and a C1 function h(-,X) : O\ —> E^iO) depending continuously on X, where E^O) is the completion of E2 in the Hg-norm, with 0 < 6 < 1 as in (2.27), such that (1) ft(0,A) = 0, ^(0,A) = 0, (2) the set Mx = {(x,y) £H\x£Ox,y
= h(x,X) G E%(9)} ,
(2.31)
called the center manifolds, are locally invariant for (2.25), i.e. for each u0 £ Mx, ux(t,u0)£Mx,
V0 0 and k\ > 0 with kx depending on (x\(0),yx(0)) such that
WyxW-hixxW^WnKkxe-P**.
36
Bifurcation Theory and Applications
If we only consider the existence of the local center manifold, then the conditions in (2.29) can be modified in the following fashion. Let the operator L\ = Lx ® £-2 > a n d £2 be decomposed into / fX
rX
ffi
rX
*-2 — *"21 ^ ^22» £/ 2 = &21 © - ^ 2 2 '
. ^
= ^0^2,
(2.32)
dim .E^i = d i m ^ < 00, , ^2i : ^2i ~* ^2i
are
invariant (i = 1,2),
such that at A = Ao [ eigenvalues of C\ : E^ —> E± have zero real parts, < eigenvalues of £31 : -^21 ~* ^21 have positive real parts, and ( eigenvalues of ££2 : -^22 ~* ^22
nave
(2.33)
negative real parts.
Theorem 2.13 Assume (2.26)—(2.28), (2.32) and (2.33). conclusions (1) and (2) in Theorem 2.12 hold true.
Then the
We now consider the case where L\ : Hi —> H generates a strongly continuous semi-group of linear operators. Assume that G(-,X) : H —> H are Cr (r > 1) bounded continuously depending on A s R 1 , and G(u, A) = o(||u|| JJ),
VAGK1.
(2.34)
Also, we assume that for any uo S Hi, the equation (2.25) has a unique solution u{t) £ Hi for all t>0. Theorem 2.14 Assume that L\ : Hi —» H generates a strongly continuous semi-group of linear operators, and the conditions (2.26), (2.28), (2.29) and (2.34) hold true. If the strongly continuous semi-group S\(t) generated by £2 satisfies
II5A(t)II < Kxe-a^,
(2.35)
for some constants K\ >l,a\>0, then there exists a C1 center manifold function h{-, A) : B\ —> E2 such that the assertions (1)—(3) in Theorem 2.12 hold, in which the center manifold M\ is replaced by
MA = {(x,y) £H\x£BxcEi,y
= h(x,y) G E$ C H] .
37
Introduction to Dynamic Bifurcation
Remark 2.3 Theorem 2.14 can be proved by using the same method as in [Henry, 1981], which is useful in the dynamic bifurcation of nonlinear wave equations. 2.4.3
Construction of center manifolds
In later discussions, it is necessary to show how the center manifold is constructed. In this subsection we give a sketch of the proof for Theorem 2.12. Let pE:Ef^> [0,1] be a C°° cut-off function defined by H V ;
f 1
if ||x|| < e,
\ 0
if ||x|| > 2s,
for some e > 0. We denote CQ>l{E$,E${6)) = {h : E$ -> E%{0) \ h{0) = 0, h is Lipschitz} . We need to find a function h € C°'x(^E^, E^B)) satisfying h(-)= [
J — oo
e-c^Pe(x(T,-))G2(x(T,-),h(x(r,.)))dT
(2.36)
where x(t,xo) is a solution of the ordinary differential equation
{
f/r
-=C>x
+ Mx)Gl{xMx),X),
(237)
x(0) = xo,
Then, it is easy to see that the function y(t,h(x0)) = h(x(t,x0)) = / J—oo
= /
J—oo
= f
J — oo
e-c*Tpe-G2(x(T,x(t,xo)),h)dT e-c*Tpe-G2(x(t
+ T,x0),h(x(t + T,x0)))dT
ec^t-^pE(x(T,xo))G2(x(T,xo),h(x(T,xo)))dT
satisfies the equation f ^ = £$V + \ 1,(0) = h(x0).
Pe(x(t,xo))G2(x(t,xo),y),
38
Bifurcation Theory and Applications
Thus, (x(t,xo),h(x(t,xo))) is a local solution of (2.30), and the manifold given by (2.31) is locally invariant for (2.25). Hence, the existence of the center manifold of (2.25) is referred to as the existence of the fixed point of the mapping
F : C°>l{El,E${6)) —> C 0 ' 1 ^ , ££(
