Elements of Applied Bifurcation Theory

Applied Mathematical Sciences Volume 112 Editors J.E. Marsden L. Sirovich Advisors S. Antman J.K. Hale P. Holmes T. Kambe J. Keller B.J. Matkowsky C.S. Peskin Springer New York Berlin Heidelberg Barcelona Hong Kong London Milan Paris Singapore Tokyo Applied Mathematical Sciences 1. John: Partial Differential Equations, 4th ed. 2. Sirovich: Techniques of Asymptotic Analysis. 3. Hale: Theory of Functional Differential Equations, 2nd ed. 4. Perots: Combinatorial Methods. 5. von Mises/Friedrichs: Fluid Dynamics. 6. Freiberger/Grenander: A Short Course in Computational Probability and Statistics. 7. Pipkin: Lectures on Viscoelasticity Theory. 8. Giacaglia: Perturbation Methods in Non-linear Systems. 9. Friedrichs: Spectral Theory of Operators in Hilbert Space. 10. Stroud: Numerical Quadrature and Solution of Ordinary Differential Equations. 11. Wolovich: Linear Multivariable Systems. 12. Berkovilz: Optimal Control Theory. 13. Bluman/Cole: Similarity Methods for Differential Equations. 14. Yoshizawa: Stability Theory and the Existence of Periodic Solution and Almost Periodic Solutions. 15. Braun: Differential Equations and Their Applications, 3rd ed. 16. Lefscbetz: Applications of Algebraic Topology. 17. Collatz/Wetterling: Optimization Problems. 18. Grenander: Pattern Synthesis; Lectures in Pattern Theory, Vol. I. 19. Marsden/McCracken: Hopf Bifurcation and Its Applications. 20. Driver: Ordinary and Delay Differential Equations. 21. Courant/Friedrichs: Supersonic Flow and Shock Waves. 22. Rouche/Habets/Laloy: Stability Theory by Liapunov's Direct Method. 23. Lamperti: Stochastic Processes: A Survey of the Mathematical Theory. 24. Grenander: Pattern Analysis: Lectures in Pattern Theory, Vol. IL 25. Davies: Integral Transforms and Their Applications, 2nd ed. 26. Kushner/Clark: Stochastic Approximation Methods for Constrained and Unconstrained Systems. 27. de Boor: A Practical Guide to Splines. 28. Keilson: Markov Chain Models—Rarity and Exponentiality. 29. de Veubeke: A Course in Elasticity. 30. Sniatycki: Geometric Quantization and Quantum Mechanics. 31. Reid: Sturmian Theory for Ordinary Differential Equations. 32. Meis/Markowitz: Numerical Solution of Partial Differential Equations. 33. Grenander: Regular Structures: Lectures in Pattern Theory, Vol. III. 34. Kevorkian/Cole: Perturbation Methods in Applied Mathematics. 35. Carr: Applications of Centre Manifold Theory. 36. Bengtsson/Ghil/Kallen: Dynamic Meteorology: Data Assimilation Methods. 37. Saperstone: Semidynamical Systems in Infinite Dimensional Spaces. 38. Lichtenberg/Lieberman: Regular and Chaotic Dynamics, 2nd ed. 39. Piccini/Stampacchia/Vidossich: Ordinary Differential Equations in R". 40. Naylor/Sell: Linear Operator Theory in Engineering and Science. 41. Sparrow: The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors. 42. Guckenheimer/Holmes: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. 43. Ockendon/Taylor: Inviscid Fluid Flows. 44. Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equations. 45. Glashojf/Gustafson: Linear Operations and Approximation: An Introduction to the Theoretical Analysis and Numerical Treatment of Semi-Infinite Programs. 46. Wilcox: Scattering Theory for Diffraction Gratings. 47. Hale et al: An Introduction to Infinite Dimensional Dynamical Systems—Geometric Theory. 48. Murray: Asymptotic Analysis. 49. Ladyzhenskaya: The Boundary-Value Problems of Mathematical Physics. 50. Wilcox: Sound Propagation in Stratified Fluids. 51. Golubitsky/Schaejfer: Bifurcation and Groups in Bifurcation Theory, Vol. I. 52. Chipol: Variational Inequaliries and Flow in Porous Media. 53. Majda: Compressible Fluid Flow and System of Conservation Laws in Several Space Variables. 54. Wasow: Linear Turning Point Theory. 55. Yosida: Operational Calculus: A Theory of Hyperfunctions. 56. Chang/Howes: Nonlinear Singular Perturbation Phenomena: Theory and Applicafions. 57. Reinhardt: Analysis of Approximafion Methods for Differential and Integral Equations. 58. Dwoyer/Hussaini/Voigt (eds): Theoredcal Approaches to Turbulence. 59. Sanders/Verhulst: Averaging Methods in Nonlinear Dynamical Systems. 60. Ghil/Childress: Topics in Geophysical Dynamics: Atmospheric Dynamics, Dynamo Theory and Climate Dynamics. (continued following index) Yuri A. Kuznetsov Elements of Applied Bifurcation Theory Second 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 142292 Pushchino, Moscow Region Russia Editors I. Karatzas Departments of Mathematics and Statistics Columbia University New York, NY 10027, USA M. Yor CNRS, Laboratoire de Probabilites Universite Pierre et Marie Curie 4, Place Jussieu, Tour 56 F-75252 Paris Cedex 05, France Mathematics Subject Classification (2000): 34C23, 37Gxx, 37M20, 3704 Library of Congress Cataloging-in-Publication Data Kuznet sov, lU. A. (iDril Aleksandrovich) Elements of applied bifurcation theoryA'uri A. Kuznetsov.—2nd ed. p. cffl. — (Applied mathematical sciences; 112) Includes bibliographical references and index. ISBN 0-387-98382-1 (hardcover: alk. paper) 1. Bifurcation theory. I. Title. II. Series: Applied mathematical sciences (Springer-Verlag New York, Inc.); v. 112. QA1.A647 vol, 112 1998 [QA380] 515'.352—dc21 97-39336 © 1998, 1995 Springer-Verlag New York, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the pubUsher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), 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. ISBN 0-387-98382-1 SPIN 10841547 Springer-Verlag New York Berlin Heidelberg A member of BertelsmannSpringer Science+Business Media GmbH To my family Preface to the Second Edition The favorable reaction to the first edition of this book confirmed that the publication of such an application-oriented text on bifurcation theory of dynamical systems was well timed. The selected topics indeed cover major practical issues of applying the bifurcation theory to finite-dimensional problems. This new edition preserves the structure of the first edition while updating the context to incorporate recent theoretical developments, in particular, new and improved numerical methods for bifurcation analysis. The treatment of some topics has been clarified. Major additions can be summarized as follows: In Chapter 3, an elementary proof of the topological equivalence of the original and truncated normal forms for the fold bifurcation is given. This makes the analysis of codimension-one equilibrium bifurcations of ODEs in the book complete. This chapter also includes an example of the Hopf bifurcation analysis in a planar system using MAPLE, a symbolic manipulation software. Chapter 4 includes a detailed normal form analysis of the Neimark-Sacker bifurcation in the delayed logistic map. In Chapter 5, we derive explicit formulas for the critical normal form coefficients of all codim 1 bifurcations of n-dimensional iterated maps (i.e., fold, flip, and Neimark-Sacker bifurcations). The section on homoclinic bifurcations in n-dimensional ODEs in Chapter 6 is completely rewritten and introduces the Melnikov integral that allows us to verify the regularity of the manifold splitting under parameter variations. Recently proved results on the existence of center manifolds near homoclinic bifurcations are also included. By their means the study of generic codim 1 homoclinic bifurcations in n-dimensional systems is reduced to that in some two-, three-, or four-dimensional systems. viii Preface to the Second Edition Two- and three-dimensional cases are treated in the main text, while the analysis of bifurcations in four-dimensional systems with a homoclinic orbit to a focus-focus is outlined in the new appendix. In Chapter 7, an explicit example of the “blue sky” bifurcation is discussed. Chapter 10, devoted to the numerical analysis of bifurcations, has been changed most substantially. We have introduced bordering methods to continue fold and Hopf bifurcations in two parameters. In this approach, the defining function for the bifurcation used in the minimal augmented system is computed by solving a bordered linear system. It allows for explicit computation of the gradient of this function, contrary to the approach when determinants are used as the defining functions. The main text now includes BVP methods to continue codim 1 homoclinic bifurcations in two parameters, as well as all codim 1 limit cycle bifurcations. A new appendix to this chapter provides test functions to detect all codim 2 homoclinic bifurcations involving a single homoclinic orbit to an equilibrium. The software review in Appendix 3 to this chapter is updated to present recently developed programs, including AUTO97 with HomCont, DsTool, and CONTENT providing the information on their availability by ftp. A number of misprints and minor errors have been corrected while preparing this edition. I would like to thank many colleagues who have sent comments and suggestions, including E. Doedel (Concordia University, Montreal), B. Krauskopf (VU, Amsterdam), S. van Gils (TU Twente, Enschede), B. Sandstede (WIAS, Berlin), W.-J. Beyn (Bielefeld University), F.S. Berezovskaya (Center for Ecological Problems and Forest Productivity, Moscow), E. Nikolaev and E.E. Shnoll (IMPB, Pushchino, Moscow Region), W. Langford (University of Guelph), O. Diekmann (Utrecht University), and A. Champneys (University of Bristol). I am thankful to my wife, Lioudmila, and to my daughters, Elena and Ouliana, for their understanding, support, and patience, while I was working on this book and developing the bifurcation software CONTENT. Finally, I would like to acknowledge the Research Institute for Applications of Computer Algebra (RIACA, Eindhoven) for the financial support of my work at CWI (Amsterdam) in 1995–1997. Yuri A. Kuznetsov Amsterdam September 1997 Preface to the First Edition During the last few years, several good textbooks on nonlinear dynamics have appeared for graduate students in applied mathematics. It seems, however, that the majority of such books are still too theoretically oriented and leave many practical issues unclear for people intending to apply the theory to particular research problems. This book is designed for advanced undergraduate or graduate students in mathematics who will participate in applied research. It is also addressed to professional researchers in physics, biology, engineering, and economics who use dynamical systems as modeling tools in their studies. Therefore, only a moderate mathematical background in geometry, linear algebra, analysis, and differential equations is required. A brief summary of general mathematical terms and results, which are assumed to be known in the main text, appears at the end of the book. Whenever possible, only elementary mathematical tools are used. For example, we do not try to present normal form theory in full generality, instead developing only the portion of the technique sufficient for our purposes. The book aims to provide the student (or researcher) with both a solid basis in dynamical systems theory and the necessary understanding of the approaches, methods, results, and terminology used in the modern applied mathematics literature. A key theme is that of topological equivalence and codimension, or “what one may expect to occur in the dynamics with a given number of parameters allowed to vary.” Actually, the material covered is sufficient to perform quite complex bifurcation analysis of dynamical systems arising in applications. The book examines the basic topics of bifurcation theory and could be used to compose a course on nonlin- x Preface to the First Edition ear dynamical systems or systems theory. Certain classical results, such as Andronov-Hopf and homoclinic bifurcation in two-dimensional systems, are presented in great detail, including self-contained proofs. For more complex topics of the theory, such as homoclinic bifurcations in more than two dimensions and two-parameter local bifurcations, we try to make clear the relevant geometrical ideas behind the proofs but only sketch them or, sometimes, discuss and illustrate the results but give only references of where to find the proofs. This approach, we hope, makes the book readable for a wide audience and keeps it relatively short and able to be browsed. We also present several recent theoretical results concerning, in particular, bifurcations of homoclinic orbits to nonhyperbolic equilibria and one-parameter bifurcations of limit cycles in systems with reflectional symmetry. These results are hardly covered in standard graduate-level textbooks but seem to be important in applications. In this book we try to provide the reader with explicit procedures for application of general mathematical theorems to particular research problems. Special attention is given to numerical implementation of the developed techniques. Several examples, mainly from mathematical biology, are used as illustrations. The present text originated in a graduate course on nonlinear systems taught by the author at the Politecnico di Milano in the Spring of 1991. A similar postgraduate course was given at the Centrum voor Wiskunde en Informatica (CWI, Amsterdam) in February, 1993. Many of the examples and approaches used in the book were first presented at the seminars held at the Research Computing Centre1 of the Russian Academy of Sciences (Pushchino, Moscow Region). Let us briefly characterize the content of each chapter. Chapter 1. Introduction to dynamical systems. In this chapter we introduce basic terminology. A dynamical system is defined geometrically as a family of evolution operators ϕt acting in some state space X and parametrized by continuous or discrete time t. Some examples, including symbolic dynamics, are presented. Orbits, phase portraits, and invariant sets appear before any differential equations, which are treated as one of the ways to define a dynamical system. The Smale horseshoe is used to illustrate the existence of very complex invariant sets having fractal structure. Stability criteria for the simplest invariant sets (equilibria and periodic orbits) are formulated. An example of infinite-dimensional continuous-time dynamical systems is discussed, namely, reaction-diffusion systems. Chapter 2. Topological equivalence, bifurcations, and structural stability of dynamical systems. Two dynamical systems are called topologically equivalent if their phase portraits are homeomorphic. This notion is 1 Renamed in 1992 as the Institute of Mathematical Problems of Biology (IMPB). Preface to the First Edition xi then used to define structurally stable systems and bifurcations. The topological classification of generic (hyperbolic) equilibria and fixed points of dynamical systems defined by autonomous ordinary differential equations (ODEs) and iterated maps is given, and the geometry of the phase portrait near such points is studied. A bifurcation diagram of a parameter-dependent system is introduced as a partitioning of its parameter space induced by the topological equivalence of corresponding phase portraits. We introduce the notion of codimension (codim for short) in a rather naive way as the number of conditions defining the bifurcation. Topological normal forms (universal unfoldings of nondegenerate parameter-dependent systems) for bifurcations are defined, and an example of such a normal form is demonstrated for the Hopf bifurcation. Chapter 3. One-parameter bifurcations of equilibria in continuous-time dynamical systems. Two generic codim 1 bifurcations – tangent (fold) and Andronov-Hopf – are studied in detail following the same general approach: (1) formulation of the corresponding topological normal form and analysis of its bifurcations; (2) reduction of a generic parameterdependent system to the normal form up to terms of a certain order; and (3) demonstration that higher-order terms do not affect the local bifurcation diagram. Step 2 (finite normalization) is performed by means of polynomial changes of variables with unknown coefficients that are then fixed at particular values to simplify the equations. Relevant normal form and nondegeneracy (genericity) conditions for a bifurcation appear naturally at this step. An example of the Hopf bifurcation in a predator-prey system is analyzed. Chapter 4. One-parameter bifurcations of fixed points in discrete-time dynamical systems. The approach formulated in Chapter 3 is applied to study tangent (fold), flip (period-doubling), and Hopf (NeimarkSacker) bifurcations of discrete-time dynamical systems. For the NeimarkSacker bifurcation, as is known, a normal form so obtained captures only the appearance of a closed invariant curve but does not describe the orbit structure on this curve. Feigenbaum’s universality in the cascade of period doublings is explained geometrically using saddle properties of the perioddoubling map in an appropriate function space. Chapter 5. Bifurcations of equilibria and periodic orbits in ndimensional dynamical systems. This chapter explains how the results on codim 1 bifurcations from the two previous chapters can be applied to multidimensional systems. A geometrical construction is presented upon which a proof of the Center Manifold Theorem is based. Explicit formulas are derived for the quadratic coefficients of the Taylor approximations to the center manifold for all codim 1 bifurcations in both continuous and discrete time. An example is discussed where the linear approximation of the center manifold leads to the wrong stability analysis of an equilibrium. We present in detail a projection method for center manifold computation that avoids the transformation of the system into its eigenbasis. Using this xii Preface to the First Edition method, we derive a compact formula to determine the direction of a Hopf bifurcation in multidimensional systems. Finally, we consider a reactiondiffusion system on an interval to illustrate the necessary modifications of the technique to handle the Hopf bifurcation in some infinite-dimensional systems. Chapter 6. Bifurcations of orbits homoclinic and heteroclinic to hyperbolic equilibria. This chapter is devoted to the generation of periodic orbits via homoclinic bifurcations. A theorem due to Andronov and Leontovich describing homoclinic bifurcation in planar continuous-time systems is formulated. A simple proof is given which uses a constructive C 1 -linearization of a system near its saddle point. All codim 1 bifurcations of homoclinic orbits to saddle and saddle-focus equilibrium points in threedimensional ODEs are then studied. The relevant theorems by Shil’nikov are formulated together with the main geometrical constructions involved in their proofs. The role of the orientability of invariant manifolds is emphasized. Generalizations to more dimensions are also discussed. An application of Shil’nikov’s results to nerve impulse modeling is given. Chapter 7. Other one-parameter bifurcations in continuoustime dynamical systems. This chapter treats some bifurcations of homoclinic orbits to nonhyperbolic equilibrium points, including the case of several homoclinic orbits to a saddle-saddle point, which provides one of the simplest mechanisms for the generation of an infinite number of periodic orbits. Bifurcations leading to a change in the rotation number on an invariant torus and some other global bifurcations are also reviewed. All codim 1 bifurcations of equilibria and limit cycles in Z2 -symmetric systems are described together with their normal forms. Chapter 8. Two-parameter bifurcations of equilibria in continuous-time dynamical systems. One-dimensional manifolds in the direct product of phase and parameter spaces corresponding to the tangent and Hopf bifurcations are defined and used to specify all possible codim 2 bifurcations of equilibria in generic continuous-time systems. Topological normal forms are presented and discussed in detail for the cusp, BogdanovTakens, and generalized Andronov-Hopf (Bautin) bifurcations. An example of a two-parameter analysis of Bazykin’s predator-prey model is considered in detail. Approximating symmetric normal forms for zero-Hopf and HopfHopf bifurcations are derived and studied, and their relationship with the original problems is discussed. Explicit formulas for the critical normal form coefficients are given for the majority of the codim 2 cases. Chapter 9. Two-parameter bifurcations of fixed points in discrete-time dynamical systems. A list of all possible codim 2 bifurcations of fixed points in generic discrete-time systems is presented. Topological normal forms are obtained for the cusp and degenerate flip bifurcations with explicit formulas for their coefficients. An approximate normal form is presented for the Neimark-Sacker bifurcation with cubic degeneracy (Chenciner bifurcation). Approximating normal forms are expressed Preface to the First Edition xiii in terms of continuous-time planar dynamical systems for all strong resonances (1:1, 1:2, 1:3, and 1:4). The Taylor coefficients of these continuoustime systems are explicitly given in terms of those of the maps in question. A periodically forced predator-prey model is used to illustrate resonant phenomena. Chapter 10. Numerical analysis of bifurcations. This final chapter deals with numerical analysis of bifurcations, which in most cases is the only tool to attack real problems. Numerical procedures are presented for the location and stability analysis of equilibria and the local approximation of their invariant manifolds as well as methods for the location of limit cycles (including orthogonal collocation). Several methods are discussed for equilibrium continuation and detection of codim 1 bifurcations based on predictor-corrector schemes. Numerical methods for continuation and analysis of homoclinic bifurcations are also formulated. Each chapter contains exercises, and we have provided hints for the most difficult of them. The references and comments to the literature are summarized at the end of each chapter as separate bibliographical notes. The aim of these notes is mainly to provide a reader with information on further reading. The end of a theorem’s proof (or its absence) is marked by the symbol 2, while that of a remark (example) is denoted by ♦ (3), respectively. As is clear from this Preface, there are many important issues this book does not touch. In fact, we study only the first bifurcations on a route to chaos and try to avoid the detailed treatment of chaotic dynamics, which requires more sophisticated mathematical tools. We do not consider important classes of dynamical systems such as Hamiltonian systems (e.g., KAM-theory and Melnikov methods are left outside the scope of this book). Only introductory information is provided on bifurcations in systems with symmetries. The list of omissions can easily be extended. Nevertheless, we hope the reader will find the book useful, especially as an interface between undergraduate and postgraduate studies. This book would have never appeared without the encouragement and help from many friends and colleagues to whom I am very much indebted. The idea of such an application-oriented book on bifurcations emerged in discussions and joint work with A.M. Molchanov, A.D. Bazykin, E.E. Shnol, and A.I. Khibnik at the former Research Computing Centre of the USSR Academy of Sciences (Pushchino). S. Rinaldi asked me to prepare and give a course on nonlinear systems at the Politecnico di Milano that would be useful for applied scientists and engineers. O. Diekmann (CWI, Amsterdam) was the first to propose the conversion of these brief lecture notes into a book. He also commented on some of the chapters and gave friendly support during the whole project. S. van Gils (TU Twente, Enschede) read the manuscript and gave some very useful suggestions that allowed me to improve the content and style. I am particularly thankful to A.R. Champneys xiv Preface to the First Edition of the University of Bristol, who reviewed the whole text and not only corrected the language but also proposed many improvements in the selection and presentation of the material. Certain topics have been discussed with J. Sanders (VU/RIACA/CWI, Amsterdam), B. Werner (University of Hamburg), E. Nikolaev (IMPB, Pushchino), E. Doedel (Concordia University, Montreal), B. Sandstede (IAAS, Berlin), M. Kirkilonis (CWI, Amsterdam), J. de Vries (CWI, Amsterdam), and others, whom I would like to thank. Of course, the responsibility for all remaining mistakes is mine. I would also like to thank A. Heck (CAN, Amsterdam) and V.V. Levitin (IMPB, Pushchino/CWI, Amsterdam) for computer assistance. Finally, I thank the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO) for providing financial support during my stay at CWI, Amsterdam. Yuri A. Kuznetsov Amsterdam December 1994 Contents Preface to the Second Edition vii Preface to the First Edition ix 1 Introduction to Dynamical Systems 1.1 Definition of a dynamical system . . . . . . . . . . . . . . . 1.1.1 State space . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Time . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Evolution operator . . . . . . . . . . . . . . . . . . . 1.1.4 Definition of a dynamical system . . . . . . . . . . . 1.2 Orbits and phase portraits . . . . . . . . . . . . . . . . . . . 1.3 Invariant sets . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Definition and types . . . . . . . . . . . . . . . . . . 1.3.2 Example 1.9 (Smale horseshoe) . . . . . . . . . . . . 1.3.3 Stability of invariant sets . . . . . . . . . . . . . . . 1.4 Differential equations and dynamical systems . . . . . . . . 1.5 Poincaré maps . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Time-shift maps . . . . . . . . . . . . . . . . . . . . 1.5.2 Poincaré map and stability of cycles . . . . . . . . . 1.5.3 Poincaré map for periodically forced systems . . . . 1.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Appendix 1: Infinite-dimensional dynamical systems defined by reaction-diffusion equations . . . . . . . . . . . . . . . . 1.8 Appendix 2: Bibliographical notes . . . . . . . . . . . . . . 1 1 2 5 5 7 8 11 11 12 16 18 23 24 25 30 31 33 37 xvi Preface to the First Edition 2 Topological Equivalence, Bifurcations, and Structural Stability of Dynamical Systems 2.1 Equivalence of dynamical systems . . . . . . . . . . . . . . . 2.2 Topological classification of generic equilibria and fixed points 2.2.1 Hyperbolic equilibria in continuous-time systems . . 2.2.2 Hyperbolic fixed points in discrete-time systems . . 2.2.3 Hyperbolic limit cycles . . . . . . . . . . . . . . . . . 2.3 Bifurcations and bifurcation diagrams . . . . . . . . . . . . 2.4 Topological normal forms for bifurcations . . . . . . . . . . 2.5 Structural stability . . . . . . . . . . . . . . . . . . . . . . . 2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Appendix: Bibliographical notes . . . . . . . . . . . . . . . . 3 One-Parameter Bifurcations of Equilibria in Continuous-Time Dynamical Systems 3.1 Simplest bifurcation conditions . . . . . . 3.2 The normal form of the fold bifurcation . 3.3 Generic fold bifurcation . . . . . . . . . . 3.4 The normal form of the Hopf bifurcation . 3.5 Generic Hopf bifurcation . . . . . . . . . . 3.6 Exercises . . . . . . . . . . . . . . . . . . 3.7 Appendix 1: Proof of Lemma 3.2 . . . . . 3.8 Appendix 2: Bibliographical notes . . . . 39 39 46 46 49 54 57 63 68 73 76 . . . . . . . . . . . . . . . . . . . . . . . . 79 . 79 . 80 . 83 . 86 . 91 . 104 . 108 . 111 4 One-Parameter Bifurcations of Fixed Points in Discrete-Time Dynamical Systems 4.1 Simplest bifurcation conditions . . . . . . . . . . . . . 4.2 The normal form of the fold bifurcation . . . . . . . . 4.3 Generic fold bifurcation . . . . . . . . . . . . . . . . . 4.4 The normal form of the flip bifurcation . . . . . . . . . 4.5 Generic flip bifurcation . . . . . . . . . . . . . . . . . . 4.6 The “normal form” of the Neimark-Sacker bifurcation 4.7 Generic Neimark-Sacker bifurcation . . . . . . . . . . . 4.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 Appendix 1: Feigenbaum’s universality . . . . . . . . . 4.10 Appendix 2: Proof of Lemma 4.3 . . . . . . . . . . . . 4.11 Appendix 3: Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 113 114 116 119 121 125 129 138 139 143 149 5 Bifurcations of Equilibria and Periodic Orbits in n-Dimensional Dynamical Systems 5.1 Center manifold theorems . . . . . . . . . . . . . . . 5.1.1 Center manifolds in continuous-time systems 5.1.2 Center manifolds in discrete-time systems . . 5.2 Center manifolds in parameter-dependent systems . 5.3 Bifurcations of limit cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 151 152 156 157 162 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preface to the First Edition 5.4 5.5 5.6 5.7 xvii Computation of center manifolds . . . . . . . . . . . . . . . 5.4.1 Quadratic approximation to center manifolds in eigenbasis . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Projection method for center manifold computation Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 1: Hopf bifurcation in reaction-diffusion systems on the interval with Dirichlet boundary conditions . . . . . Appendix 2: Bibliographical notes . . . . . . . . . . . . . . 6 Bifurcations of Orbits Homoclinic and Heteroclinic to Hyperbolic Equilibria 6.1 Homoclinic and heteroclinic orbits . . . . . . . . . . . 6.2 Andronov-Leontovich theorem . . . . . . . . . . . . . . 6.3 Homoclinic bifurcations in three-dimensional systems: Shil’nikov theorems . . . . . . . . . . . . . . . . . . . . 6.4 Homoclinic bifurcations in n-dimensional systems . . . 6.4.1 Regular homoclinic orbits: Melnikov integral . 6.4.2 Homoclinic center manifolds . . . . . . . . . . . 6.4.3 Generic homoclinic bifurcations in Rn . . . . . 6.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Appendix 1: Focus-focus homoclinic bifurcation in four-dimensional systems . . . . . . . . . . . . . . . 6.7 Appendix 2: Bibliographical notes . . . . . . . . . . . 165 165 171 186 189 193 195 . . . 195 . . . 200 . . . . . . . . . . . . . . . . . . 213 228 229 232 236 238 . . . 241 . . . 247 7 Other One-Parameter Bifurcations in Continuous-Time Dynamical Systems 249 7.1 Codim 1 bifurcations of homoclinic orbits to nonhyperbolic equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 7.1.1 Saddle-node homoclinic bifurcation on the plane . . 250 7.1.2 Saddle-node and saddle-saddle homoclinic bifurcations in R3 . . . . . . . . . . . . . . . . . . . 253 7.2 “Exotic” bifurcations . . . . . . . . . . . . . . . . . . . . . . 262 7.2.1 Nontransversal homoclinic orbit to a hyperbolic cycle 263 7.2.2 Homoclinic orbits to a nonhyperbolic limit cycle . . 263 7.3 Bifurcations on invariant tori . . . . . . . . . . . . . . . . . 267 7.3.1 Reduction to a Poincaré map . . . . . . . . . . . . . 267 7.3.2 Rotation number and orbit structure . . . . . . . . . 269 7.3.3 Structural stability and bifurcations . . . . . . . . . 270 7.3.4 Phase locking near a Neimark-Sacker bifurcation: Arnold tongues . . . . . . . . . . . . . . . . . . . . . 272 7.4 Bifurcations in symmetric systems . . . . . . . . . . . . . . 276 7.4.1 General properties of symmetric systems . . . . . . . 276 7.4.2 Z2 -equivariant systems . . . . . . . . . . . . . . . . . 278 7.4.3 Codim 1 bifurcations of equilibria in Z2 -equivariant systems . . . . . . . . . . . . . . . . . . . . . . . . . 280 xviii Preface to the First Edition 7.4.4 7.5 7.6 Codim 1 bifurcations of cycles in Z2 -equivariant systems . . . . . . . . . . . . . . . 283 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 Appendix 1: Bibliographical notes . . . . . . . . . . . . . . 290 8 Two-Parameter Bifurcations of Equilibria in Continuous-Time Dynamical Systems 8.1 List of codim 2 bifurcations of equilibria . . . . . . . . . . . 8.1.1 Codim 1 bifurcation curves . . . . . . . . . . . . . . 8.1.2 Codim 2 bifurcation points . . . . . . . . . . . . . . 8.2 Cusp bifurcation . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Normal form derivation . . . . . . . . . . . . . . . . 8.2.2 Bifurcation diagram of the normal form . . . . . . . 8.2.3 Effect of higher-order terms . . . . . . . . . . . . . . 8.3 Bautin (generalized Hopf) bifurcation . . . . . . . . . . . . 8.3.1 Normal form derivation . . . . . . . . . . . . . . . . 8.3.2 Bifurcation diagram of the normal form . . . . . . . 8.3.3 Effect of higher-order terms . . . . . . . . . . . . . . 8.4 Bogdanov-Takens (double-zero) bifurcation . . . . . . . . . 8.4.1 Normal form derivation . . . . . . . . . . . . . . . . 8.4.2 Bifurcation diagram of the normal form . . . . . . . 8.4.3 Effect of higher-order terms . . . . . . . . . . . . . . 8.5 Fold-Hopf (zero-pair) bifurcation . . . . . . . . . . . . . . . 8.5.1 Derivation of the normal form . . . . . . . . . . . . . 8.5.2 Bifurcation diagram of the truncated normal form . 8.5.3 Effect of higher-order terms . . . . . . . . . . . . . . 8.6 Hopf-Hopf bifurcation . . . . . . . . . . . . . . . . . . . . . 8.6.1 Derivation of the normal form . . . . . . . . . . . . . 8.6.2 Bifurcation diagram of the truncated normal form . 8.6.3 Effect of higher-order terms . . . . . . . . . . . . . . 8.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8 Appendix 1: Limit cycles and homoclinic orbits of Bogdanov normal form . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.9 Appendix 2: Bibliographical notes . . . . . . . . . . . . . . 9 Two-Parameter Bifurcations of Fixed Points in Discrete-Time Dynamical Systems 9.1 List of codim 2 bifurcations of fixed points . . . . . . 9.2 Cusp bifurcation . . . . . . . . . . . . . . . . . . . . 9.3 Generalized flip bifurcation . . . . . . . . . . . . . . 9.4 Chenciner (generalized Neimark-Sacker) bifurcation . 9.5 Strong resonances . . . . . . . . . . . . . . . . . . . . 9.5.1 Approximation by a flow . . . . . . . . . . . 9.5.2 1:1 resonance . . . . . . . . . . . . . . . . . . 9.5.3 1:2 resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 294 294 297 301 301 303 305 307 307 312 313 314 314 321 324 330 330 337 342 349 349 356 366 369 382 390 393 393 397 400 404 408 408 410 415 Preface to the First Edition . . . . . 428 435 446 457 460 10 Numerical Analysis of Bifurcations 10.1 Numerical analysis at fixed parameter values . . . . . . . . 10.1.1 Equilibrium location . . . . . . . . . . . . . . . . . . 10.1.2 Modified Newton’s methods . . . . . . . . . . . . . . 10.1.3 Equilibrium analysis . . . . . . . . . . . . . . . . . . 10.1.4 Location of limit cycles . . . . . . . . . . . . . . . . 10.2 One-parameter bifurcation analysis . . . . . . . . . . . . . . 10.2.1 Continuation of equilibria and cycles . . . . . . . . . 10.2.2 Detection and location of codim 1 bifurcations . . . 10.2.3 Analysis of codim 1 bifurcations . . . . . . . . . . . 10.2.4 Branching points . . . . . . . . . . . . . . . . . . . . 10.3 Two-parameter bifurcation analysis . . . . . . . . . . . . . . 10.3.1 Continuation of codim 1 bifurcations of equilibria and fixed points . . . . . . . . . . . . . . . . . . . . 10.3.2 Continuation of codim 1 limit cycle bifurcations . . 10.3.3 Continuation of codim 1 homoclinic orbits . . . . . . 10.3.4 Detection and location of codim 2 bifurcations . . . 10.4 Continuation strategy . . . . . . . . . . . . . . . . . . . . . 10.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6 Appendix 1: Convergence theorems for Newton methods . . 10.7 Appendix 2: Detection of codim 2 homoclinic bifurcations . 10.7.1 Singularities detectable via eigenvalues . . . . . . . . 10.7.2 Orbit and inclination flips . . . . . . . . . . . . . . . 10.7.3 Singularities along saddle-node homoclinic curves . . 10.8 Appendix 3: Bibliographical notes . . . . . . . . . . . . . . 463 464 464 466 469 472 478 479 484 488 495 501 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 . . . . . . . . . . . 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 . . . . . . . . . 9.6 9.7 9.8 9.5.4 1:3 resonance . . . . . . . . . 9.5.5 1:4 resonance . . . . . . . . . Codim 2 bifurcations of limit cycles . Exercises . . . . . . . . . . . . . . . Appendix 1: Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix 501 507 510 514 515 517 525 526 527 529 534 535 . . . . . . . . 541 541 541 543 544 . . . . . . . . . . . . . . 545 546 546 547 547 548 549 xx Preface to the First Edition A.3 Geometry . . . . A.3.1 Sets . . . A.3.2 Maps . . A.3.3 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550 550 551 551 References 553 Index 577