Applied Mathematical Sciences Volume 112
Editors J.E. Marsden L. Sirovich F. John (deceased)
Advisors M. OhiI J.K. Hale T. Kambe J. Keller K. Kirchgassner B.J. Matkowsky C.S. Peskin J.T. Stuart
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. Percus: 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. Giacoglia: 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. II. Wolovich: Linear Multivariable Systems. 12. Berkovitz: Optimal Control Theory. 13. BlumaniCole: 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. Le{schetz: Applications of Algebraic TopolOgy. 17. Collatz/Wetterling: Optimization Problems. 18. Grenander: Pattern Synthesis: Lectures in Pattern Theory. Vol. I. 19. MarsdeniMcCracken: HopfBifurcation 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 Uapunov's Direct Method. 23. Lamperti: Stochastic Processes: A Survey of the Mathematical Theory. 24. Grenander: Pattern Analysis: Lectures in Pattern Theory. Vol. II. 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. Shiatycki: 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. m. 34. KevorkianiCole: Perturbation Methods in Applied Mathematics. 35. Carr: Applications of Centre Manifold Theory. 36. BengtssoniGhillK .. II.n: Dynamic Meteorology: Data Assimilation Methods. 37. Saperstone: Semidynamical Systems in Infinite Dimensional Spaces. 38. Lichtenberg/Lieberman: Regular and Chaotic Dynamics. 2nd ed. 39. Piccini/StampacchiaiVitiossich: Ordinary Differential Equations in JR" . 40. Naylor/Sell: Linear Operator Theory in Engineering and Science. 41. Spa"ow: The Lorenz Equations: Bifurcations. Chaos. and Strange Attractors. 42. Guckenheimer/Holmes: Nonlinear Oscillations. Dynamical Systems and Bifurcations of Vector'Fields. 43. OckendoniTaylor: Inviscid Fluid Flows. 44. Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equations. 45. GlashoffiGustafson: 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.
(continued following index)
Yuri A. Kuznetsov
Elements of Applied Bifurcation Theory
With 232 Illustrations
Springer Science+Business Media, LLC
Yuri A. Kuznetsov Centrum voor Wiskunde en Informatica Kruislan 413 1098 SJ Amsterdam The Netherlands and Institute of Mathematical Problems of Biology Russian Academy of Sciences Pushchino, Moscow Region 142292 Russia
Editors J.E. Marsden Department of
Mathematics University of California Berkeley, CA 94720 USA
L. Sirovich Division of
Applied Mathematics Brown University Providence, RI02912 USA
Mathematics Subject Classification (1991): 34A47, 35B32, 58F14, 58F39
Library of Congress Cataloging-in-Publication Data Kuznetsov, fu. A. (furii Aleksandrovich)
Elements of applied bifurcation theory / Yuri A. Kuznetsov. p. cm. - (Applied mathematical sciences ; 112)
Includes bibliographical references and index.
1. Bifurcation theory. I. Title. II. Series: Applied Mathematical Sciences Springer Science+Business Media, LLC. v. 112. QA380.K891995 94-41847 515'.352-dc20
Printed on acid-free paper.
© 1995 Springer Science+Business Media New York Originally published by Springer-Verlag New York, Inc. in 1995 Softcover reprint of the hardcover 1st edition 1995
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987654321
ISBN 978-1-4757-2423-3
ISBN 978-1-4757-2423-3 ISBN 978-1-4757-2421-9 (eBook)DOI 10.1007/978-1-4757-2421-9
Tomyfamily
Preface
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 that 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 modem 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 nonlinear dynamical systems or system 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
viii Preface
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 Centre' 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 ({Jt 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 invcpiant sets having fractal structure. Stability criteria for the simplest invariant sets (equilibria and periodic orbits) are formulated. An example of infinite-dimensional continuoustime 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 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
iRenamed in 1992 as the Institute of Mathematical Problems of Biology (IMPB).
Preface ix
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 systems. Two generic codim 1 bifurcations - tangent (fold) and Andronov-Hopfare 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 parameter-dependent system to the normal form up to terms of a certain order; (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 which 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 systems. The approach formulated in Chapter 3 is applied to study tangent (fold), flip (period-doubling), and Hopf (Neimark-Sacker) bifurcations of discrete-time dynamical systems. For the Neimark-Sacker 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 period-doubling map in an appropriate function space.
Chapter 5. Bifurcations of equilibria and periodic orbits in n-dimensional 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 I 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 method, we derive a compact formula to determine the direction of a Hopf bifurcation in multidimensional systems. Finally, we consider a reaction-diffusion 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 I-linearization of a system near its saddle point. All codim 1 bifurcations of homoclinic orbits to saddle and saddle-focus equilibrium points in three-dimensional ODEs are then studied. The relevant theorems
x Preface
by Shil'nikov are fonnulated 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 continuous-time 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 nonnal fonns.
Chapter 8. Two-parameter bifurcations of equilibria in continuous-time 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 nonnal fonns are presented and discussed in detail for the cusp, Bogdanov-Takens, 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 nonnal fonns for zero-Hopf and Hopf-Hopf bifurcations are derived and studied, and their relationship with the original problems is discussed. Explicit fonnulas for the critical nonnal fonn coefficients are given for the majority of the codim 2 cases.
Chapter 9. Two-parameter bifurcations of fixed points in discrete-time systems. A list of all possible codim 2 bifurcations of fixed points in generic discretetime systems is presented. Topological nonnal fonns are obtained for the cusp and degenerate flip bifurcations with explicit fonnulas for their coefficients. An approximate nonnal fonn is presented for the Neimark-Sacker bifurcation with cubic degeneracy (Chenciner bifurcation). Approximating nonnal fonns are expressed in tenns of continuous-time planar dynamical systems for all strong resonances (1:1, 1:2, 1:3, and 1:4). The Taylor coefficients of these continuous-time systems are explicitly given in tenns of those ofthe 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 fonnulated.
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
Preface xi
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 D, while that ofaremark (example) is denoted by <:>( <», 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 (for example, 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 appeared in discussions and joint work with A. M. Molchanov, A. D. Bazykin, E. E. Shnol and A.1. 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 of the University of Bristol, who reviewd 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 (VUIRIACNCWI, 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 should 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 various 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 vii
1 Introduction to Dynamical Systems 1 1.1 Definition of dynamical system 1 1.2 Orbits and phase portraits . . . 8 1.3 Invariant sets . . . . . . . . . . 10 1.4 Differential equations and dynamical systems 17 1.5 Poincare maps . . . . . . . . . . . . . . . . 22 1.6 Exercises................... 29 1.7 Appendix 1: Infinite-dimensional dynamical systems defined by
reaction-diffusion equations . . . . 30 1.8 Appendix 2: Bibliographical notes . . . . . . . . . . . . . . .. 34
2 Topological Equivalence, Bifurcations, and Structural Stability of Dynamical Systems 36 2.1 Equivalence of dynamical systems . . . . . . . . . . . . . . . 36 2.2 Topological classification of generic equilibria and fixed points 42 2.3 Bifurcations and bifurcation diagrams . . 53 2.4 Topological normal forms for bifurcations 59 2.5 Structural stability . . . . . . . . 63 2.6 Exercises............. 67 2.7 Appendix: Bibliographical notes . 71
3 One-Parameter Bifurcations of Equilibria in Continuous-Time Systems 73 3.1 Simplest bifurcation conditions .... 73 3.2 The normal form of the fold bifurcation 74 3.3 Fold bifurcation theorem. . . . . . . . 77 3.4 The normal form of the Hopf bifurcation. 79 3.5 Hopf bifurcation theorem .... 84 3.6 Exercises.............. 97 3.7 Appendix 1: Proof of Lemma 3.2 . 99 3.8 Appendix 2: Bibliographical notes . 101
xiv Contents
4 One-Parameter Bifurcations of Fixed Points in Discrete-Time Systems 103 4.1 Simplest bifurcation conditions .... 103 4.2 The nonnal fonn ofthe fold bifurcation 104 4.3 Fold bifurcation theorem . . . . . . . . 106 4.4 The nonnal fonn ofthe flip bifurcation 108 4.5 Flip bifurcation theorem . . . . . . . . 111 4.6 The "nonnal fonn" of the Neimark-Sacker bifurcation. 114 4.7 Neimark-Sacker bifurcation theorem. . 118 4.8 Exercises................ 126 4.9 Appendix 1: Feigenbaum's universality 127 4.10 Appendix 2: Proof of Lemma 4.3 . 131 4.11 Appendix 3: Bibliographical notes. . . 136
5 Bifurcations of Equilibria and Periodic Orbits in n-Dimensional Systems 138 5.1 Center manifold theorems . . . . . . . . . . . . . 138 5.2 Center manifolds in parameter-dependent systems. 144 5.3 Bifurcations of limit cycles. . . . 148 5.4 Computation of center manifolds. . . . . . . . . . 151 5.5 Exercises...................... 166 5.6 Appendix 1: Hopf bifurcation in reaction-diffusion systems on
the interval with Dirichlet boundary conditions 168 5.7 Appendix 2: Bibliographical notes. . . . . . . . . . . . . .. 176
6 Bifurcations of Orbits Homoclinic and Heteroclinic to Hyperbolic Equilibria 178 6.1 Homoclinic and heteroclinic orbits . . . . . . . . . . . 178 6.2 Andronov-Leontovich theorem. . . . . . . . . . . . . 183 6.3 Homoclinic bifurcations in three-dimensional systems:
Shil'nikov theorems ............... . 6.4 Homoclinic bifurcations in n-dimensional systems. 6.5 Exercises.............. 6.6 Appendix 1: Bibliographical notes . . . . . . . . .
193 208 210 212
7 Other One-Parameter Bifurcations in Continuous-Time Systems 214 7.1 Codim 1 bifurcations of homoclinic orbits to nonhyperbolic
equilibria. . . . . . . . . . . 214 7.2 "Exotic" bifurcations. . . . . . . . 227 7.3 Bifurcations on invariant tori . . . . 229 7.4 Bifurcations in symmetric systems. 236 7.5 Exercises............. 248 7.6 Appendix 1: Bibliographical notes . 250
Contents xv
8 Two-Parameter Bifurcations of Equilibria in Continuous-Time Dynamical Systems 252 8.1 List of codim 2 bifurcations of equilibria. 252 8.2 Cusp bifurcation . . . . . . . . . . . . . 259 8.3 Bautin (generalized Hopf) bifurcation . . 265 8.4 Bogdanov-Takens (double-zero) bifurcation 272 8.5 Fold-Hopf (zero-pair) bifurcation 287 8.6 Hopf-Hopf bifurcation . . . . . . . . . . . 305 8.7 Exercises.................. 324 8.8 Appendix 1: Limit cycles and homoclinic orbits of the
Bogdanov-Takens normal form .. 336 8.9 Appendix 2: Bibliographical notes . . . . . . . . . . . . 345
9 Two-Parameter Bifurcations of Fixed Points in Discrete-Time Dynamical Systems 347 9.1 List of codim 2 bifurcations of fixed points 347 9.2 Cusp bifurcation . . . . . . . . . . . . . . 351 9.3 Generalized flip bifurcation . . . . . . . . 354 9.4 Chenciner (generalized Neimark-Sacker) bifurcation 357 9.5 Strong resonances . . . . . . . . . 362 9.6 Codim 2 bifurcations of limit cycles 399 9.7 Exercises.............. 408 9.8 Appendix 1: Bibliographical notes. 412
10 Numerical Analysis of Bifurcations 414 10.1 Numerical analysis at fixed parameter values 415 10.2 One-parameter bifurcation analysis 429 10.3 1Wo-parameter bifurcation analysis 449 10.4 Continuation strategy ....... 455 10.5 Exercises . . . . . . . . . . . . . . 456 10.6 Appendix 1: Convergence theorems for Newton methods 464 10.7 Appendix 2: Numerical analysis of homoclinic bifurcations 465 10.8 Appendix 3: Bibliographical notes. . . . . . . . . 472
A Basic Notions from Algebra, Analysis, and Geometry 477 A.1 Algebra . 477 A.2 Analysis . 482 A.3 Geometry 485
References 487
Index 503
