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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