Elements of Applied Bifurcation Theory

Transcription

1 Yuri A. Kuznetsov Elements of Applied Bifurcation Theory Third Edition With 251 Illustrations Springer

2 Yuri A. Kuznetsov Department of Mathematics Utrecht University Budapestlaan CD Utrecht The Netherlands and Institute of Mathematical Problems of Biology Russian Academy of Sciences Pushchino, Moscow Region Russia Editors: S.S. Antman Department of Mathematics and Institute for Physical Science and Technology University of Maryland College Park, MD USA ssa@math.umd.edu J.E. Marsden Control and Dynamical Systems, California Institute of Technology Pasadena, CA USA L. Sirovich Division of Applied Mathematics Brown University Providence, RI 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 ISBN (ebook) DOI / 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 SPIN springeronline.com

3 1 Introduction to Dynamical Systems Definition of a dynamical system State space Time Evolution operator Definition of a dynamical system Orbits and phase portraits Invariant sets Definition and types Smale horseshoe Stability of invariant sets Differential equations and dynamical systems Poincare maps Time-shift maps Poincare map and stability of cycles Poincare map for periodically forced systems Exercises Appendix A: Infinite-dimensional dynamical systems defined by reaction-diffusion equations Appendix B: Bibliographical notes Topological Equivalence, Bifurcations, and Structural Stability of Dynamical Systems Equivalence of dynamical systems Topological classification of generic equilibria and fixed points Hyperbolic equilibria in continuous-time systems Hyperbolic fixed points in discrete-time systems Hyperbolic limit cycles Bifurcations and bifurcation diagrams Topological normal forms for bifurcations Structural stability... 67

4 XVIII 2.6 Exercises Appendix: Bibliographical notes One-Parameter Bifurcations of Equilibria in Continuous- Time Dynamical Systems Simplest bifurcation conditions The normal form of the fold bifurcation Generic fold bifurcation The normal form of the Hopf bifurcation Generic Hopf bifurcation Exercises Appendix A: Proof of Lemma Appendix B: Poincare normal forms Appendix C: Bibliographical notes One-Parameter Bifurcations of Fixed Points in Discrete-Time Dynamical Systems Simplest bifurcation conditions The normal form of the fold bifurcation Generic fold bifurcation The normal form of the fiip bifurcation Generic flip bifurcation The "normal form" of the Neimark-Sacker bifurcation Generic Neimark-Sacker bifurcation Exercises Appendix A: Feigenbaum's universality Appendix B: Proof of Lemma Appendix C: Bibliographical notes Bifurcations of Equilibria and Periodic Orbits in n-dimensional Dynamical Systems Center manifold theorems Center manifolds in continuous-time systems Center manifolds in discrete-time systems Center manifolds in parameter-dependent systems Bifurcations of limit cycles Computation of center manifolds Restricted normalized equations for ODEs Restricted normalizedequations for maps Exercises Appendix A: Hopf bifurcation in reaction-diffusion systems Appendix B: Bibliographical notes

5 XIX 6 Bifurcations of Orbits Homoclinic and Heteroclinic to Hyperbolic Equilibria Homoclinie and heteroclinic orbits Andronov-Leontovieh theorem Homoclinie bifurcations in three-dimensional systems: Shil'nikov theorems Homoclinie bifurcations in n-dimensional systems Regular homoclinie orbits: Melnikov integral Homoclinie center manifolds Generie homoclinie bifurcations in ]Rn Exercises Appendix A: Focus-focus homoclinie bifurcation in four-dimensional systems Appendix B: Bibliographieal notes Other One-Parameter Bifurcations in Continuous-Time Dynamical Systems Codim 1 bifurcations of homoclinie orbits to nonhyperbolic equilibria Saddle-node homoclinie bifurcation on the plane Saddle-node and saddle-saddle homoclinie bifurcations in]r Bifurcations of orbits homoclinie to limit cycles Nontransversal homoclinie orbit to a hyperbolic cycle Homoclinie orbits to a nonhyperbolic limit cycle Bifurcations on invariant tori Reduction to a Poincare map Rotation number and orbit structure Structural stability and bifurcations Phase locking near a Neimark-Sacker bifurcation: Arnold tongues Bifurcations in symmetrie systems General properties of symmetrie systems Z2-equivariant systems Codim 1 bifurcations of equilibria in Z2-equivariant systems Codim 1 bifurcations of cycles in Z2-equivariant systems Exercises Appendix: Bibliographieal notes Two-Parameter Bifurcations of Equilibria in Continuous- Time Dynamical Systems List of codim 2 bifurcations of equilibria Codim 1 bifurcation curves Codim 2 bifurcation points

6 XX 8.2 Cusp bifurcation Normal form derivation Bifurcation diagram of the normal form Effect of higher-order terms Bautin (generalized Hopf) bifurcation Normal form derivation Bifurcation diagram of the normal form Effect of higher-order terms Bogdanov-Takens (double-zero) bifurcation Normal form derivation Bifurcation diagram of the normal form Effect of higher-order terms Fold-Hopf bifurcation Derivation of the normal form Bifurcation diagram of the truncated normal form Effect of higher-order terms Hopf-Hopf bifurcation Derivation of the normal form Bifurcation diagram of the truncated normal form Effect of higher-order terms Critical normal forms for n-dimensional systems The method Cusp bifurcation Bautin bifurcation Bogdanov-Takens bifurcation Fold-Hopf bifurcation Hopf-Hopf bifurcation Exercises Appendix A: Limit cycles and hornoclinie orbits of Bogdanov normal form Appendix B: Bibliographical notes Two-Parameter Bifurcations of Fixed Points in Discrete-Time Dynamical Systems List of codim 2 bifurcations of fixed points Cusp bifurcation Generalized flip bifurcation Chenciner (generalized Neimark-Sacker) bifurcation Strong resonances Approximation by a flow :1 resonance :2 resonance :3 resonance :4 resonance Fold-flip bifurcation

7 XXI 9.7 Critical normal forms for n-dimensional maps Cusp Generalized flip Chenciner bifurcation Resonance 1: Resonance 1: Resonance 1:3..., Resonance 1: Fold-flip Codim 2 bifurcations of limit cycles Exercises Appendix: Bibliographical notes Numerical Analysis of Bifurcations Numerical analysis at fixed parameter values Equilibrium location Modified Newton's methods Equilibrium analysis Location of limit cycles One-parameter bifurcation analysis Continuation of equilibria and cycles Detection and location of codim 1 bifurcations Analysis of codim 1 bifurcations Branching points..., Two-parameter bifurcation analysis Continuation of codim 1 bifurcations of equilibria and fixed points Continuation of codim 1 limit cycle bifurcations Continuation of codim 1 homoclinic orbits Detection, location, and analysis of codim 2 bifurcations Continuation strategy Exercises Appendix A: Convergence theorems for Newton methods Appendix B: Bialternate matrix product Appendix C: Detection of codim 2 homoclinic bifurcations Singularities detectable via eigenvalues Orbit and inclination flips Singularities along saddle-node homoclinic curves Appendix D: Bibliographical notes A Basic Notions from Algebra, Analysis, and Geometry A.1 Algebra A.1.1 Matrices A.1.2 Vector spaces and linear transformations A.1.3 Eigenvectors and eigenvalues

8 XXII A.1.4 Invariant subspaces, generalized eigenvectors, and Jordan normal form A.1.5 Fredholm Alternative Theorem A.1.6 Groups A.2 Analysis A.2.1 Implicit and Inverse Function Theorems A.2.2 Taylor expansion A.2.3 Metric, normed, and other spaces A.3 Geometry A.3.1 Sets A.3.2 Maps A.3.3 Manifolds References Index