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

by Haragus, Mariana; Iooss, Gerard

• ISBN13: 9780857291110
• ISBN10: 0857291114
• Format: Paperback
• Copyright: 2010-12-03
• Publisher: Springer Verlag

Summary

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.

Table of Contents

Elementary Bifurcations	p. 1
Bifurcations in Dimension 1	p. 1
Saddle-Node Bifurcation	p. 2
Pitchfork Bifurcation	p. 5
Bifurcations in Dimension 2	p. 8
Hopf Bifurcation	p. 9
Example: Homogeneous Brusselator	p. 18
Hopf Bifurcation with SO(2) Symmetry	p. 22
Steady Bifurcation with O(2) Symmetry	p. 24
Center Manifolds	p. 29
Notations	p. 29
Local Center Manifolds	p. 30
Hypotheses	p. 30
Main Result	p. 34
Checking Hypothesis 2.7	p. 36
Examples	p. 38
Particular Cases and Extensions	p. 46
Parameter-Dependent Center Manifolds	p. 46
Nonautonomous Center Manifolds	p. 51
Symmetries and Reversibility	p. 53
Empty Unstable Spectrum	p. 58
Further Examples and Exercises	p. 59
A Fourth Order ODE	p. 60
Burgers Model	p. 63
Swift-Hohenberg Equation	p. 70
Brusselator Model	p. 81
Elliptic PDE in a Strip	p. 89
Normal Forms	p. 93
Main Theorem	p. 93
Proof of Theorem 1.2	p. 95
Examples in Dimension 2: i¿, 02	p. 99
Examples in Dimension 3: 0(i¿), 03	p. 104
Examples in Dimension 4: (i¿1) (i¿2), (i¿)2, 02(i¿), 0202	p. 106
Parameter-Dependent Normal Forms	p. 109
Main Result	p. 109
Linear Normal Forms	p. 111
Derivation of the Parameter-Dependent Normal Form	p. 112
Example: 02 Normal Form with Parameters	p. 114
Symmetries and Reversibility	p. 116
Equivariant Vector Fields	p. 117
Reversible Vector Fields	p. 118
Example: van der Pol System	p. 120
Normal Forms for Reduced Systems on Center Manifolds	p. 122
Computation of Center Manifolds and Normal Forms	p. 122
Example 1: Hopf Bifurcation	p. 124
Example 2: Hopf Bifurcations with Symmetries	p. 127
Example 3: Takens-Bogdanov Bifurcation	p. 134
Example 4: (i¿1)(i¿2) bifurcation	p. 139
Further Normal Forms	p. 144
Time-Periodic Normal Forms	p. 144
Example: Periodically Forced Hopf Bifurcation	p. 147
Normal Forms for Analytic Vector Fields	p. 152
Reversible Bifurcations	p. 157
Dimension 2	p. 157
Reversible Takens-Bogdanov Bifurcation 02+	p. 160
Reversible Takens-Bogdanov Bifurcation 02-	p. 172
Dimension 3	p. 179
Reversible 03+ Bifurcation	p. 180
Reversible 03- Bifurcation (Elements)	p. 191
Reversible 002 Bifurcation (Elements)	p. 193
Reversible 0(i¿) Bifurcation (Elements)	p. 195
Dimension 4	p. 197
Reversible 02+(i¿) Bifurcation	p. 198
Reversible 02-(i¿) Bifurcation (Elements)	p. 210
Reversible (i¿)2 Bifurcation (1-1 resonance)	p. 214
Reversible (i¿1)(i¿2) Bifurcation (Elements)	p. 226
Reversible 04+ Bifurcation (Elements)	p. 229
Reversible 0202 Bifurcation with SO(2) Symmetry	p. 234
Applications	p. 239
Hydrodynamic Instabilities	p. 239
Hydrodynamic Problem	p. 239
Couette-Taylor Problem	p. 244
Bénard-Rayleigh Convection Problem	p. 249
Existence of Traveling Waves	p. 258
Gravity-Capillary Water-Waves	p. 259
Almost-Planar Waves in Reaction-Diffusion Systems	p. 270
Waves in Lattices	p. 275
Appendix	p. 279
Elements of Functional Analysis	p. 279
Bounded and Closed Operators	p. 279
Resolvent and Spectrum	p. 280
Compact Operators and Operators with Compact Resolvent	p. 282
Adjoint Operator	p. 283
Fredholm Operators	p. 284
Basic Sobolev Spaces	p. 284
Center Manifolds	p. 287
Proof of Theorem 2.9 (Center Manifolds)	p. 287
Proof of Theorem 2.17 (Semilinear Case)	p. 293
Proof of Theorem 3.9 (Nonautonomous Vector Fields)	p. 298
Proof of Theorem 3.13 (Equivariant Systems)	p. 299
Proof of Theorem 3.22 (Empty Unstable Spectrum)	p. 300
Normal Forms	p. 301
Proof of Lemma 1.13 (03 Normal Form)	p. 302
Proof of Lemma 1.17 ((i¿)2 Normal Form)	p. 303
Proof of Lemma 1.18 (02(i¿) Normal Form)	p. 305
Proof of Lemma 1.19 (0202 Normal Form)	p. 307
Proof of Theorem 2.2 (Perturbed Normal Forms)	p. 308
Reversible Bifurcations	p. 310
03+ Normal Form in Infinite Dimensions	p. 310
(i¿)2 Normal Form in Infinite Dimensions	p. 315
References	p. 321
Index	p. 327
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