Yuri A. Kuznetsov Elements of Applied Bifurcation Theory Third Edition With 251 Illustrations Springer
Yuri A. Kuznetsov Department of Mathematics Utrecht University Budapestlaan 6 3584 CD Utrecht The Netherlands and Institute of Mathematical Problems of Biology Russian Academy of Sciences 142290 Pushchino, Moscow Region Russia Editors: S.S. Antman Department of Mathematics and Institute for Physical Science and Technology University of Maryland College Park, MD 20742-4015 USA ssa@math.umd.edu J.E. Marsden Control and Dynamical Systems, 107-81 California Institute of Technology Pasadena, CA 91125 USA marsden @cds.caltech.edu L. Sirovich Division of Applied Mathematics Brown University Providence, RI 02912 USA chico@camelot.mssm.edu Mathematics Subject Classification (2000): 34C23, 37Gxx, 37M20, 3704 Library of Congress Cataloging-in-Publication Data KuznefSov, lü. A. (iori! Aleksandrovich) Elements of applied bifurcation theorylyuri A. Kuznetsov.-3rd ed. [on file] ISBN 978-1-4419-1951-9 ISBN 978-1-4757-3978-7 (ebook) DOI 10.1007/978-1-4757-3978-7 Printed on acid-free paper. 2004, 1998, 1995 Springer Science+Business Media New Y ork Originally published by Springer-Verlag New York, LLC in 2004 Softcover reprint of the hardcover 3rd edition 2004 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher Springer Science+Business Media, LLC, except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. 9 8 7 6 5 4 3 2 1 SPIN 10952337 springeronline.com
1 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 2 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 fiip 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 normalizedequations 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 Homoclinie and heteroclinic orbits... 195 6.2 Andronov-Leontovieh theorem... 200 6.3 Homoclinie bifurcations in three-dimensional systems: Shil'nikov theorems... 213 6.4 Homoclinie bifurcations in n-dimensional systems... 228 6.4.1 Regular homoclinie orbits: Melnikov integral... 228 6.4.2 Homoclinie center manifolds... 232 6.4.3 Generie homoclinie bifurcations in ]Rn... 235 6.5 Exercises... 237 6.6 Appendix A: Focus-focus homoclinie bifurcation in four-dimensional systems... 240 6.7 Appendix B: Bibliographieal notes... 245 7 Other One-Parameter Bifurcations in Continuous-Time Dynamical Systems.... 249 7.1 Codim 1 bifurcations of homoclinie orbits to nonhyperbolic equilibria... 249 7.1.1 Saddle-node homoclinie bifurcation on the plane... 250 7.1.2 Saddle-node and saddle-saddle homoclinie bifurcations in]r3... 253 7.2 Bifurcations of orbits homoclinie to limit cycles... 262 7.2.1 Nontransversal homoclinie orbit to a hyperbolic cycle.. 262 7.2.2 Homoclinie 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 symmetrie systems... 278 7.4.1 General properties of symmetrie 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: Bibliographieal 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 hornoclinie 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.1 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.1.5 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