Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimensional Dynamical Systems / Edition 1

by Mariana Haragus, Gérard Iooss

Overview

An extension of different lectures given by the authors, Local Bifurcations, Center Manifolds, and Normal Forms in Infinite Dimensional Dynamical Systems provides the reader with a comprehensive overview of these topics.

Starting with the simplest bifurcation problems arising for ordinary differential equations in one- and two-dimensions, this book describes several tools from the theory of infinite dimensional dynamical systems, allowing the reader to treat more complicated bifurcation problems, such as bifurcations arising in partial differential equations. Attention is restricted to the study of local bifurcations with a focus upon the center manifold reduction and the normal form theory; two methods that have been widely used during the last decades.

Through use of step-by-step examples and exercises, a number of possible applications are illustrated, and allow the less familiar reader to use this reduction method by checking some clear assumptions. Written by recognised experts in the field of center manifold and normal form theory this book provides a much-needed graduate level text on bifurcation theory, center manifolds and normal form theory. It will appeal to graduate students and researchers working in dynamical system theory.

Product Details

ISBN-13:	9780857291110
Publisher:	Springer London
Publication date:	12/08/2010
Series:	Universitext
Edition description:	2011
Pages:	329
Product dimensions:	6.10(w) x 9.20(h) x 0.90(d)

Table of Contents

1 Elementary Bifurcations 1

1.1 Bifurcations in Dimension 1 1

1.1.1 Saddle-Node Bifurcation 2

1.1.2 Pitchfork Bifurcation 5

1.2 Bifurcations in Dimension 2 8

1.2.1 Hopf Bifurcation 9

1.2.2 Example: Homogeneous Brusselator 18

1.2.3 Hopf Bifurcation with SO(2) Symmetry 22

1.2.4 Steady Bifurcation with O(2) Symmetry 24

2 Center Manifolds 29

2.1 Notations 29

2.2 Local Center Manifolds 30

2.2.1 Hypotheses 30

2.2.2 Main Result 34

2.2.3 Checking Hypothesis 2.7 36

2.2.4 Examples 38

2.3 Particular Cases and Extensions 46

2.3.1 Parameter-Dependent Center Manifolds 46

2.3.2 Nonautonomous Center Manifolds 51

2.3.3 Symmetries and Reversibility 53

2.3.4 Empty Unstable Spectrum 58

2.4 Further Examples and Exercises 59

2.4.1 A Fourth Order ODE 60

2.4.2 Burgers Model 63

2.4.3 Swift-Hohenberg Equation 70

2.4.4 Brusselator Model 81

2.4.5 Elliptic PDE in a Strip 89

3 Normal Forms 93

3.1 Main Theorem 93

3.1.1 Proof of Theorem 1.2 95

3.1.2 Examples in Dimension 2: iω, 0² 99

3.1.3 Examples in Dimension 3: 0(iω), 0³ 104

3.1.4 Examples in Dimension 4: (iω₁) (iω₂), (iω)², 0²(iω), 0²0² 106

3.2 Parameter-Dependent Normal Forms 109

3.2.1 Main Result 109

3.2.2 Linear Normal Forms 111

3.2.3 Derivation of the Parameter-Dependent Normal Form 112

3.2.4 Example: 0² Normal Form with Parameters 114

3.3 Symmetries and Reversibility 116

3.3.1 Equivariant Vector Fields 117

3.3.2 Reversible Vector Fields 118

3.3.3 Example: van der Pol System 120

3.4 Normal Forms for Reduced Systems on Center Manifolds 122

3.4.1 Computation of Center Manifolds and Normal Forms 122

3.4.2 Example 1: Hopf Bifurcation 124

3.4.3 Example 2: Hopf Bifurcations with Symmetries 127

3.4.4 Example 3: Takens-Bogdanov Bifurcation 134

3.4.5 Example 4: (iω₁)(iω₂) bifurcation 139

3.5 Further Normal Forms 144

3.5.1 Time-Periodic Normal Forms 144

3.5.2 Example: Periodically Forced Hopf Bifurcation 147

3.5.3 Normal Forms for Analytic Vector Fields 152

4 Reversible Bifurcations 157

4.1 Dimension 2 157

4.1.1 Reversible Takens-Bogdanov Bifurcation 0²⁺ 160

4.1.2 Reversible Takens-Bogdanov Bifurcation 0^2- 172

4.2 Dimension 3 179

4.2.1 Reversible 0³⁺ Bifurcation 180

4.2.2 Reversible 0^3- Bifurcation (Elements) 191

4.2.3 Reversible 00² Bifurcation (Elements) 193

4.2.4 Reversible 0(iω) Bifurcation (Elements) 195

4.3 Dimension 4 197

4.3.1 Reversible 0²⁺(iω) Bifurcation 198

4.3.2 Reversible 0^2-(iω) Bifurcation (Elements) 210

4.3.3 Reversible (iω)² Bifurcation (1-1 resonance) 214

4.3.4 Reversible (iω₁)(iω₂) Bifurcation (Elements) 226

4.3.5 Reversible 0⁴⁺ Bifurcation (Elements) 229

4.3.6 Reversible 0²0² Bifurcation with SO(2) Symmetry 234

5 Applications 239

5.1 Hydrodynamic Instabilities 239

5.1.1 Hydrodynamic Problem 239

5.1.2 Couette-Taylor Problem 244

5.1.3 Bénard-Rayleigh Convection Problem 249

5.2 Existence of Traveling Waves 258

5.2.1 Gravity-Capillary Water-Waves 259

5.2.2 Almost-Planar Waves in Reaction-Diffusion Systems 270

5.2.3 Waves in Lattices 275

Appendix 279

A Elements of Functional Analysis 279

A.1 Bounded and Closed Operators 279

A.2 Resolvent and Spectrum 280

A.3 Compact Operators and Operators with Compact Resolvent 282

A.4 Adjoint Operator 283

A.5 Fredholm Operators 284

A.6 Basic Sobolev Spaces 284

B Center Manifolds 287

B.1 Proof of Theorem 2.9 (Center Manifolds) 287

B.2 Proof of Theorem 2.17 (Semilinear Case) 293

B.3 Proof of Theorem 3.9 (Nonautonomous Vector Fields) 298

B.4 Proof of Theorem 3.13 (Equivariant Systems) 299

B.5 Proof of Theorem 3.22 (Empty Unstable Spectrum) 300

C Normal Forms 301

C.1 Proof of Lemma 1.13 (0³ Normal Form) 302

C.2 Proof of Lemma 1.17 ((iω)² Normal Form) 303

C.3 Proof of Lemma 1.18 (0²(iω) Normal Form) 305

C.4 Proof of Lemma 1.19 (0²0² Normal Form) 307

C.5 Proof of Theorem 2.2 (Perturbed Normal Forms) 308

D Reversible Bifurcations 310

D.1 0³⁺ Normal Form in Infinite Dimensions 310

D.2 (iω)² Normal Form in Infinite Dimensions 315

References 321

Index 327

Editorial Reviews

From the reviews:

'This book relies on versions of the center manifold theorem that apply to infinite-dimensional dynamical systems...Chapter 4 of the book is distinctive in its presentation of normal forms for bifurcations in ‘reversible’ systems. These are systems in which there is a symmetry that reverses the orientation of time. When this symmetry is a reflection, it leads to systems that have large families of periodic orbits because forward and backward trajectories that start on the subspace must meet if they return to this subspace. This is an intricate subject, and this book makes it much more accessible than ever before. As for much of Iooss’ work throughout his career, this book gives many concrete examples of problems described by PDEs with an excellent balance between theory and applications of that theory' (SIAM Review, December 2011)

“This book relies on versions of the center manifold theorem that apply to infinite-dimensional dynamical systems. … This is an intricate subject, and this book makes it much more accessible than ever before. As for much of Iooss’ work throughout his career, this book gives many concrete examples of problems described by PDEs with an excellent balance between theory and applications of that theory.”­­­ (John Guckenheimer, SIAM Review, Vol. 53 (4), 2011)

From the Publisher