Series Editor: Leon 0.

Chua

BIFURCATION THEORY
AND APPLICATIONS
Tian Ma
Sichuan university, China

Shouhong Wang
Indiana university, USA

> World Scientific

NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • H O N G K O N G • TAIPEI • CHENNAI
 Contents

Preface vii

1. Introduction to Steady State Bifurcation Theory 1

1.1 Implicit Function Theorem 1
1.2 Basics of Topological Degree Theory 2
1.2.1 Brouwer degree 2
1.2.2 Basic theorems of Brouwer degree 4
1.2.3 Leray-Schauder degree 5
1.2.4 Indices of isolated singularities 7
1.3 Lyapunov-Schmidt Method 8
1.3.1 Preliminaries 8
1.3.2 Lyapunov-Schmidt procedure 9
1.3.3 Normalization 12
1.4 Krasnosel'ski Bifurcation Theorems 13
1.4.1 Bifurcation from eigenvalues with odd multiplicity . 13
1.4.2 Krasnosel'ski theorem for potential operators . . . . 14
1.5 Rabinowitz Global Bifurcation Theorem 17
1.6 Notes 19

2. Introduction to Dynamic Bifurcation 21

2.1 Motivation 21
2.2 Semi-groups of Linear Operators 23
2.2.1 Introduction 23
2.2.2 Strongly continuous semi-groups 25
2.2.3 Sectorial operators and analytic semi-groups 26
2.2.4 Powers of linear operators 28
x Bifurcation Theory and Applications

2.3 Dissipative Dynamical Systems 29

2.4 Center Manifold Theorems 32
2.4.1 Center and stable manifolds in Rn 32
2.4.2 Center manifolds for infinite dimensional systems . . 34
2.4.3 Construction of center manifolds 37
2.5 Hopf Bifurcation 38
2.6 Notes 40

3. Reduction Procedures and Stability 41

3.1 Spectrum Theory of Linear Completely Continuous Fields . 41
3.1.1 Eigenvalues of linear completely continuous fields . . 41
3.1.2 Spectral theorems 44
3.1.3 Asymptotic properties of eigenvalues 50
3.1.4 Generic properties 53
3.2 Reduction Methods 56
3.2.1 Reduction procedures 56
3.2.2 Morse index of nondegenerate singular points . . . . 62
3.3 Asymptotic Stability at Critical States 66
3.3.1 Introduction to the Lyapunov stability 66
3.3.2 Finite dimensional cases 67
3.3.3 An alternative principle for stability 70
3.3.4 Dimension reduction 72
3.4 Notes 74

4. 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

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5.1 Introduction 105

5.1.1 Pendulum in a symmetric magnetic field 105
5.1.2 Business cycles for Kaldor's model 110
5.1.3 Basic principle of attractor bifurcation 112
5.2 Attractor Bifurcation 114
5.2.1 Main theorems 114
5.2.2 Stability of attractors 116
5.2.3 Proof of Theorems 5.2 and 5.3 119
5.2.4 Structure of bifurcated attractors 123
5.2.5 Generalized Hopf bifurcation 127
5.3 Invariant Closed Manifolds 129
5.3.1 Hyperbolic invariant manifolds 129
5.3.2 S 1 attractor bifurcation 132
5.4 Stability of Dynamic Bifurcation 138
5.5 Notes 149

6. Dynamic Bifurcation Theory: Infinite Dimensional Case 151

6.1 Attractor Bifurcation 152
6.1.1 Equations with first-order in time 152
6.1.2 Equations with second-order in time 154
6.2 Bifurcation from Simple Eigenvalues 160
6.2.1 Structure of dynamic bifurcation 160
6.2.2 Saddle-node bifurcation 163
6.3 Bifurcation from Eigenvalues with Multiplicity Two .... 165
6.3.1 An index formula 165
6.3.2 Main theorems 169
6.3.3 Proof of main theorems 172
6.3.4 Case where k > 3 184
6.3.5 Bifurcation to periodic solutions 184
6.4 Stability for Perturbed Systems 188
6.4.1 General case 188
6.4.2 Perturbation at simple eigenvalues 191
6.5 Notes 194

7. Bifurcations for Nonlinear Elliptic Equations 197

7.1 Preliminaries 197
7.1.1 Sobolev spaces 197
7.1.2 Regularity estimates 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
m
8.3 Singularity Sphere in 5 -Attractors 251
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 267
9.1.1 Set-up 267
9.1.2 Symmetric case 268
9.1.3 General case 271
9.1.4 S1-invariant sets 273
9.2 Cahn-Hillard Equation 275
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9.2.1 Set-up 275

9.2.2 Neumann boundary condition 276
9.2.3 Periodic boundary condition 286
9.2.4 Saddle-node bifurcation 290
9.3 Complex Ginzburg-Landau Equation 291
9.3.1 Set-up 291
9.3.2 Dirichlet boundary condition 293
9.3.3 Periodic boundary condition 296
9.4 Ginzburg-Landau Equations of Superconductivity 297
9.4.1 The model 297
9.4.2 Attractor bifurcation 302
9.4.3 Physical remarks 315
9.5 Wave Equations 322
9.5.1 Wave equations with damping 322
9.5.2 System of wave equations 324
9.6 Notes 325

10. Fluid Dynamics 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