Yuri A. Kuznetsov Elements of Applied Bifurcation Theory Third Edition With 251 Illustrations Springer
Introduction to Dynamical Systems 1 1.1 Definition of a dynamical system 1 1.1.1 State space 1 1.1.2 Time 5 1.1.3 Evolution operator 5 1.1.4 Definition of a dynamical system 7 1.2 Orbits and phase portraits 8 1.3 Invariant sets 10 1.3.1 Definition and types 10 1.3.2 Smale horseshoe 11 1.3.3 Stability of invariant sets 17 1.4 Differential equations and dynamical systems 18 1.5 Poincare maps 24 1.5.1 Time-shift maps 24 1.5.2 Poincare map and stability of cycles 26 1.5.3 Poincare map for periodically forced systems 31 1.6 Exercises 32 1.7 Appendix A: Infinite-dimensional dynamical systems defined by reaction-diffusion equations 33 1.8 Appendix B: Bibliographical notes 36 Topological Equivalence, Bifurcations, and Structural Stability of Dynamical Systems 39 2.1 Equivalence of dynamical systems 39 2.2 Topological classification of generic equilibria and fixed points. 45 2.2.1 Hyperbolic equilibria in continuous-time systems 46 2.2.2 Hyperbolic fixed points in discrete-time systems 49 2.2.3 Hyperbolic limit cycles 54 2.3 Bifurcations and bifurcation diagrams 57 2.4 Topological normal forms for bifurcations 63 2.5 Structural stability 67
XVIII 2.6 Exercises 72 2.7 Appendix: Bibliographical notes 75 3 One-Parameter Bifurcations of Equilibria in Continuous- Time Dynamical Systems 77 3.1 Simplest bifurcation conditions 77 3.2 The normal form of the fold bifurcation 78 3.3 Generic fold bifurcation 81 3.4 The normal form of the Hopf bifurcation 84 3.5 Generic Hopf bifurcation 89 3.6 Exercises 102 3.7 Appendix A: Proof of Lemma 3.2 106 3.8 Appendix B: Poincare normal forms 108 3.9 Appendix C: Bibliographical notes 114 4 One-Parameter Bifurcations of Fixed Points in Discrete-Time Dynamical Systems 117 4.1 Simplest bifurcation conditions 117 4.2 The normal form of the fold bifurcation 121 4.3 Generic fold bifurcation 122 4.4 The normal form of the flip bifurcation 125 4.5 Generic flip bifurcation 127 4.6 The "normal form" of the Neimark-Sacker bifurcation 131 4.7 Generic Neimark-Sacker bifurcation 136 4.8 Exercises 144 4.9 Appendix A: Feigenbaum's universality 145 4.10 Appendix B: Proof of Lemma 4.3 149 4.11 Appendix C: Bibliographical notes 154 5 Bifurcations of Equilibria and Periodic Orbits in n-dimensional Dynamical Systems 157 5.1 Center manifold theorems 157 5.1.1 Center manifolds in continuous-time systems 157 5.1.2 Center manifolds in discrete-time systems 164 5.2 Center manifolds in parameter-dependent systems 165 5.3 Bifurcations of limit cycles 170 5.4 Computation of center manifolds 172 5.4.1 Restricted normalized equations for ODEs 173 5.4.2 Restricted normalized equations for maps 182 5.5 Exercises 188 5.6 Appendix A: Hopf bifurcation in reaction-diffusion systems... 191 5.7 Appendix B: Bibliographical notes 194
XIX 6 Bifurcations of Orbits Homoclinic and Heteroclinic to Hyperbolic Equilibria 195, 6.1 Homoclinic and heteroclinic orbits 195 6.2 Andronov-Leontovich theorem 200 6.3 Homoclinic bifurcations in three-dimensional systems: Shil'nikov theorems 213 6.4 Homoclinic bifurcations in n-dimensional systems 228 6.4.1 Regular homoclinic orbits: Melnikov integral 228 6.4.2 Homoclinic center manifolds 232 6.4.3 Generic homoclinic bifurcations inr 235 6.5 Exercises 237 6.6 Appendix A: Focus-focus homoclinic bifurcation in four-dimensional systems 240 6.7 Appendix B: Bibliographical notes 245 7 Other One-Parameter Bifurcations in Continuous-Time Dynamical Systems 249 7.1 Codim 1 bifurcations of homoclinic orbits to nonhyperbolic equilibria 249 7.1.1 Saddle-node homoclinic bifurcation on the plane 250 7.1.2 Saddle-node and saddle-saddle homoclinic bifurcations in E 3 253 7.2 Bifurcations of orbits homoclinic to limit cycles 262 7.2.1 Nontransversal homoclinic orbit to a hyperbolic cycle.. 262 7.2.2 Homoclinic orbits to a nonhyperbolic limit cycle 266 7.3 Bifurcations on invariant tori 270 7.3.1 Reduction to a Poincare map 270 7.3.2 Rotation number and orbit structure 271 7.3.3 Structural stability and bifurcations 273 7.3.4 Phase locking near a Neimark-Sacker bifurcation: Arnold tongues 275 7.4 Bifurcations in symmetric systems 278 7.4.1 General properties of symmetric systems 279 7.4.2 Z2-equivariant systems 280 7.4.3 Codim 1 bifurcations of equilibria in Z2-equivariant systems 282 7.4.4 Codim 1 bifurcations of cycles in Z2-equivariant systems285 7.5 Exercises 291 7.6 Appendix: Bibliographical notes 292 8 Two-Parameter Bifurcations of Equilibria in Continuous- Time Dynamical Systems 295 8.1 List of codim 2 bifurcations of equilibria 295 8.1.1 Codim 1 bifurcation curves 296 8.1.2 Codim 2 bifurcation points 299
XX 8.2 Cusp bifurcation ' 302 8.2.1 Normal form derivation 302 8.2.2 Bifurcation diagram of the normal form 305 8.2.3 Effect of higher-order terms 307 8.3 Bautin (generalized Hopf) bifurcation 309 8.3.1 Normal form derivation 309 8.3.2 Bifurcation diagram of the normal form 313 8.3.3 Effect of higher-order terms 315 8.4 Bogdanov-Takens (double-zero) bifurcation 316 8.4.1 Normal form derivation 316 8.4.2 Bifurcation diagram of the normal form 323 8.4.3 Effect of higher-order terms 326 8.5 Fold-Hopf bifurcation 332 8.5.1 Derivation of the normal form 332 8.5.2 Bifurcation diagram of the truncated normal form 339 8.5.3 Effect of higher-order terms 345 8.6 Hopf-Hopf bifurcation 351 8.6.1 Derivation of the normal form 351 8.6.2 Bifurcation diagram of the truncated normal form 358 8.6.3 Effect of higher-order terms. 368 8.7 Critical normal forms for n-dimensional systems 370 8.7.1 The method 370 8.7.2 Cusp bifurcation 372 8.7.3 Bautin bifurcation 374 8.7.4 Bogdanov-Takens bifurcation 376 8.7.5 Fold-Hopf bifurcation 378 8.7.6 Hopf-Hopf bifurcation 382 8.8 Exercises 384 8.9 Appendix A: Limit cycles and homoclinic orbits of Bogdanov normal form 395 8.10 Appendix B: Bibliographical notes 403 9 Two-Parameter Bifurcations of Fixed Points in Discrete-Time Dynamical Systems 407 9.1 List of codim 2 bifurcations of fixed points 407 9.2 Cusp bifurcation 412 9.3 Generalized flip bifurcation 414 9.4 Chenciner (generalized Neimark-Sacker) bifurcation 418 9.5 Strong resonances 422 9.5.1 Approximation by a flow 422 9.5.2 1:1 resonance 424 9.5.3 1:2 resonance 434 9.5.4 1:3 resonance 447 9.5.5 1:4 resonance 454 9.6 Fold-flip bifurcation 466
XXI 9.7 Critical normal forms for n-dimensional maps 479 9.7.1 Cusp 480 9.7.2 Generalized flip 481 9.7.3 Chenciner bifurcation 482 9.7.4 Resonance 1:1 484 9.7.5 Resonance 1:2 485 9.7.6 Resonance 1:3 486 9.7.7 Resonance 1:4 487 9.7.8 Fold-flip 488 9.8 Codim 2 bifurcations of limit cycles 489 9.9 Exercises 498 9.10 Appendix: Bibliographical notes 502 10 Numerical Analysis of Bifurcations 505 10.1 Numerical analysis at fixed parameter values 506 10.1.1 Equilibrium location 506 10.1.2 Modified Newton's methods 508 10.1.3 Equilibrium analysis 511 10.1.4 Location of limit cycles 514 10.2 One-parameter bifurcation analysis 520 10.2.1 Continuation of equilibria and cycles 520 10.2.2 Detection and location of codim 1 bifurcations 526 10.2.3 Analysis of codim 1 bifurcations 529 10.2.4 Branching points 537 10.3 Two-parameter bifurcation analysis 543 10.3.1 Continuation of codim 1 bifurcations of equilibria and fixed points 543 10.3.2 Continuation of codim 1 limit cycle bifurcations 549 10.3.3 Continuation of codim 1 homoclinic orbits 552 10.3.4 Detection, location, and analysis of codim 2 bifurcations556 10.4 Continuation strategy 558 10.5 Exercises 559 10.6 Appendix A: Convergence theorems for Newton methods 566 10.7 Appendix B: Bialternate matrix product 568 10.8 Appendix C: Detection of codim 2 homoclinic bifurcations... 573 10.8.1 Singularities detectable via eigenvalues 574 10.8.2 Orbit and inclination flips 576 10.8.3 Singularities along saddle-node homoclinic curves 579 10.9 Appendix D: Bibliographical notes 581 A Basic Notions from Algebra, Analysis, and Geometry 587 A.I Algebra >. 587 A.1.1 Matrices 587 A. 1.2 Vector spaces and linear transformations 589 A.1.3 Eigenvectors and eigenvalues 590
XXII A.1.4 Invariant subspaces, generalized eigenvectors, and Jordan normal form 591 A.I.5 Fredholm Alternative Theorem 592 A.1.6 Groups 593 A.2 Analysis 593 A.2.1 Implicit and Inverse Function Theorems 593 A.2.2 Taylor expansion 594 A.2.3 Metric, normed, and other spaces 595 A.3 Geometry 596 A.3.1 Sets 596 A.3.2 Maps 597 A.3.3 Manifolds 597 References 599 Index 619