Applied Mathematical Sciences

Applied Mathematical Sciences Volume 51 Editors J.E. Marsden L. Sirovich F. John (deceased) Advisors S. Antman J.K. Hale P. Holmes T. Kambe J. Keller ...
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Applied Mathematical Sciences Volume 51 Editors J.E. Marsden L. Sirovich F. John (deceased) Advisors S. Antman J.K. Hale P. Holmes T. Kambe J. Keller B.J. Matkowsky C.S. Peskin

Springer Science+Business Media, LLC

Applied Mathematical Seiences 1. John: Partial Differential Equations, 4th ed. 2. Sirovich: Tecbniques of Asymptotic Analysis. 3. Haie: Tbeory ofFunctional 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 Tbeory. 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. Woiovich: Linear Multivariable Systems. 12. Berkovitz: Optimal Control Theory. 13. Biuman!Coie: Similarity Methods for Differential Equations. 14. Yoshizawa: Stability Tbeory and the Existence of Periodic Solution and Ahnost Periodic Solutions. 15. Braun: Differential Equations and Their Applications, 3rd ed. 16. Lefschetz: Applications of Algebraic Topology. 17. Collatz!Wetterling: Optimization Problems. 18. Grenander: Pattern Synthesis: Lectures in Pattern Tbeory, Vol. I. 19. Marsden!McCracken: HopfBifurcation and Its Applications. 20. Driver: Ordinary and Delay Differential Equations. 21. Courant!Friedrichs: Supersonic Flow and Shock Waves. 22. Rouche/Habets/Laioy: Stability Tbeory by Liapunov's Direct Method. 23. Lamperti: Stochastic Processes: A Survey of the Mathematical Tbeory. 24. Grenander: Pattern Analysis: Lectures in Pattern Tbeory, Vol. IL 25. Davies: Integral Transforms and Their Applications, 2nd ed. 26. Kushner!Ciark: Stochastic Approximation Methods for Constrained and Unconstrained Systems. 27. de Boor: A Practical Guide to Splines. 28. Keiison: Markov Chain Models-Rarity and Exponentiality. 29. de Veubeke: A Course in Elasticity. 30. §liatycki: Geometrie Q!I8Jl!ization and Quantum Mechanics. 31. Reid: Sturmian Tbeory for Ordinary Differential Equations. 32. Meis/Markowitz: Numerical Solution ofPartial Differential Equations. 33. Grenander: Regular Structures: Lectures in Pattern Tbeory, Vol. IIL

34. Kevorkian!Coie: Perturbation Methods in Applied Mathematics. 35. Carr: Applications of Centre Manifold Theory. 36. Bengtsson!Ghil/K(Ji/en: 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 Rn. 40. Nayior!Sell: Linear Operator Theory in Engineering and Science. 41. Sparrow: The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors. 42. Guckenheimer!Hoimes: Nonlinear Oscillations, Dynamical Systems, and Bifurcations ofVector Fields. 43. Ockendon!Tayior: Inviscid Fluid Flows. 44. Pazy: Semigroups ofLinear Operatorsand Applications to Partial Differential Equations. 45. Giashojj!Gustafson: Linear Operations and Approximation: An Introduction to the Tbeoretical Analysis and Numerical Treatment of SemiInfinite Programs. 46. Wilcox: Scattering Tbeory for Diffraction Gratings. 47. Haie et ai: An Introduction to Infinite Dimensional Dynamical Systems-Geometrie Theory. 48. Murray: Asymptotic Analysis. 49. Ladyzhenskaya: The Boundary-Value Problems of Mathematical Physics. 50. Wiicox: Sound Propagation in Stratifted Fluids. 51. Goiubitsky!Schaeffer: Bifurcation and Groups in Bifurcation Tbeory, Vol. I. 52. Chipot: Variational Inequalities and Flow in Porous Media. 53. Majda: Compressible Fluid Flow and System of Conservation Laws in Several Space Variables. 54. Wasow: Linear Turning Point Tbeory. 55. Yosida: Operational Calculus: A Tbeory of Hyperfunctions. 56. Chang!Howes: Nonlinear Singular Perturbation Phenomena: Tbeory and Applications. 57. Reinhardt: Analysis of Approximation Methods for Differential and Integral Equations. 58. Dwoyer!Hussaini/Voigt (eds): Theoretical Approaches to Turbulence. 59. Sanders!Verhuist: Averaging Methods in Nonlinear Dynamical Systems. 60. Ghil/Chiidress: Topics in Geophysical Dynamics: Atmospheric Dynamics, Dynamo Tbeory and Climate Dynamics. (continued following index)

Martin Golubitsky David G. Schaeffer

Singularities and Groups in Bifurcation Theory Volume I With 114 Illustrations


Martin Golubitsky Department of Mathematics University of Houston Houston, TX 77004 USA

David G. Schaeffer Department of Mathematics Duke University Durham, NC 27706 USA

Editors J.E. Marsden Control and Dynamical Systems, 107-81 California Institute of Technology Pasadena, CA 91125 USA

L. Sirovich Division of Applied Mathematics Brown University Providence, RI 02912 USA

Mathematics Subject Classification: 35B32, 34D05, 34D20, 34D30, 73D05, 58Fl4, 34L35, 35L67, 76L05, 57R45 Library of Congress Cataloging-in-Publication Data Golubitsky, Martin. Singularities and groups in bifurcation theory. (Applied mathematical sciences; 51) Bibliography: p. Includes index. 1. Bifurcation theory. 2. Singularity Theory. I. Schaeffer, David G. II. Title. III. Series: Applied mathematical sciences (Springer-Verlag New York Inc.); 51. QA374.G59 1984 515.3'53 84-1414 Printed on acid-free paper. © 1985 Springer Science+Business Media New York Originally published by Springer-Verlag New York, Inc. in 1985

Softcover reprint of the hardcover 1st edition 1985

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

SPIN 10782874

ISBN 978-1-4612-9533-4 ISBN 978-1-4612-5034-0 (eBook) DOI 10.1007/978-1-4612-5034-0

To To

Elizabeth and Alexander and Jennie


This book has been written in a frankly partisian spirit-we believe that singularity theory offers an extremely useful approach to bifurcation problems and we hope to convert the reader to this view. In this preface we will discuss what we feel are the strengths of the singularity theory approach. This discussion then Ieads naturally into a discussion of the contents of the book and the prerequisites for reading it. Let us emphasize that our principal contribution in this area has been to apply pre-existing techniques from singularity theory, especially unfolding theory and classification theory, to bifurcation problems. Many ofthe ideas in this part of singularity theory were originally proposed by Rene Thom; the subject was then developed rigorously by John Matherand extended by V. I. Arnold. In applying this material to bifurcation problems, we were greatly encouraged by how weil the mathematical ideas of singularity theory meshed with the questions addressed by bifurcation theory. Concerning our title, Singularities and Groups in Bifurcation Theory, it should be mentioned that the present text is the first volume in a two-volume sequence. In this volume our emphasis is on singularity theory, with group theory playing a subordinate role. In Volume II the emphasis will be more balanced. Having made these remarks, Iet us set the context for the discussion of the strengths of the singularity theory approach to bifurcation. As we use the term, bifurcation theory is the study of equations with multiple solutions. Specifically, by a bifurcation we mean a change in the number of solutions of an equation as a parameter varies. F or a wide variety of equations, including many partial differential equations, problems concerning multiple solutions can be reduced to studying how the solutions x of a single scalar equation g(x, A.) = 0 (P.l)



vary with the parameter A.. This simplification depends on a technique known as the Liapunov-Schmidt reduction. The singularity theory approach deals with equations ofthe form (P.l); i.e., with equations after the Liapunov-Schmidt reduction has been performed. We shall emphasize the qualitative properties of such equations. This emphasis is sharply focused by the notion of equivalence, which defines precisely what it means for two such equations, and their solution sets, to be qualitatively similar. The theory quickly leads one to generalize (P.l) to include k-parameter families of such equations; i.e., equations of the form G(x, A., a) = 0,


where a = (a 1 , . . . , ak) is a shorthand for k auxiliary parameters. Weshall call G an unfolding of g if for a = 0 G(x, A., 0) = g(x, A.).


Since G(x, A., a) = g(x, A.)


[G(x, A, a) - G(x, A., 0)],

we may think of G(x, A., a) as a perturbation of g(x, A.). In this volume we limit our discussion of (P.2) in the following four ways: (i) we assume that the dependence of G on x, A., and a is infinitely differentiable; (ii) we consider primarily the case where x is a scalar (one-dimensional) unknown; (iii) we work locally (i.e., in the neighborhood of some fixed point (x 0 , A. 0 )); and (iv) we discuss dynamics only in a limited way. Brief discussions of points (i) and (iii) occur later in this Preface. Concerning point (ii), in Volume II we will consider finite-dimensional systems of equations with several unknowns. Let us elaborate on point (iv). Typically, equations with multiple solutions arise in characterizing steady-state solutions of an evolution equation. Singularity theory methods are useful in finding the steady-state solutions and, in some instances, their stabilities. However, it does not seem to be possible with these methods to analyze essentially dynamic phenomena such as strange attractors. One generat strength of the singularity theory approach to bifurcation problems is easily stated-this approach unifies the treatment of many diverse problems in steady-state bifurcation. Such unification has the obvious advantage of elegance, but it also leads to economy of effort. Specifically, the same general methods used to study the most familiar problems in bifurcation theory continue to apply in a variety of nonstandard contexts. For example, whether or not g(x, A.) = 0 has a trivial solution and whether or not symmetries are present, the theoretical framework of the singularity



theory approach is the same. Also, although in this text we consider only equations having A. as a distinguished parameter, the same techniques work equally weil when all parameters are treated on the same footing. In the next few paragraphs we discuss three specific problems in bifurcation theory that are solved by the singularity theory approach; we also discuss how this information is useful for applications. The first problem, called the recognition problem, is the following: Given an equation h(x, A.) = 0, when is a second equation g(x, A.) = 0 equivalent to h(x, A.) = 0? In solving this problem, singularity theory methods produce a finite Iist of terms in the Taylor series of g such that the question of whether equivalence obtains is determined wholly by the values of the derivatives of g on this Iist-all other terms may be ignored. (Of course this Iist depends on the given function h(x, A.); moreover for certain pathological functions h(x, A.), a finite Iist does not suffice.) Regarding applications, this Iist specifies precisely the calculations which must be performed to recognize an equation of a given qualitative type. As we shall illustrate in Case Study 1, this information helps organize the computations for analyzing mathematical models. The second problern concerns perturbations of an equation g(x, A.) = 0; i.e., equations of the form g(x, A.)

+ p(x,

A.) = 0,


where p is appropriately small. Specifically, the problern is to enumerate all qualitatively different perturbations of a given equation. We will solve this problern by constructing and analyzing what is called a universal unfolding. By way of definition, a universal unfolding of g is a distinguished k-parameter family of functions, G(x, A., 0(), which satisfies (P.3) and has the following crucial property: For any small perturbation p, there is a value of il( such that g + p is equivalent to G(·, ·, 0(). Less formally, up to qualitative equivalence, G contains allsmall perturbations of g. Let us elaborate on point (i) above, the Iimitation that we consider only smooth functions of x, A., and il(. In constructing a universal unfolding of g, we will show that il( in the universal unfolding and p in (P.4) are related by a smooth transformation. Nonetheless, a great deal of nonsmooth behavior is contained in a universal unfolding. Specifically, it is rarely possible, even locally, to express the solution x of (P.2) as a smooth, or even continuous, function of A. and il(. The spirit of our approach is to work with smooth relationships between variables for as long as possible. Thus we attempt to solve (P.2) for x only after transforming the equation to a particularly tractable form; these transformations may be performed in a purely coo context. Our work with universal unfoldings has two additional benefits for applications. First, these methods often allow one to determine quasiglobal properties of a model using purely local methods (cf. point (i) above); and second, in multiparameter models, these methods impose a structure



on the physical parameter space that is useful as a guide in thinking about the problem. Both these benefits are illustrated in Case Sturlies 1 and 2. The third problern is to classify the qualitatively different equations g(x, A.) = 0 that rnay occur. This isaproblern of infinite cornplexity in that there are infinitely rnany equation types and there are equation types of arbitrarily high cornplexity. The singularity theory notion of codirnension provides a rational approach to this problern. The codimension of g is the nurnber of pararneters needed in a universal unfolding of g; this notion also provides a rough rneasure of the likelihood of an equation of a given qualitative type appearing in a rnathernatical rnodel, equations with lower codirnensions being rnore likely. Of course we do not solve the classification problern cornpletely. In this book we list all the qualitative types of equations having codirnension three or less, along with all the qualitatively different perturbations of each. (See Chapter IV.) It is possible to extend the classification to higher codirnensions, but the effort required escalates rapidly. Our list of qualitative types of equations and their perturbations includes graphs of the solution sets, which we call bifurcation diagrams. These diagrarns rnay be used in applications as follows. Consider a physical problern which depends on one or rnore auxiliary pararneters. Suppose that for various values of the pararneters one can generate representative bifurcation diagrams for the problern either by experirnent or by nurnerical solution of a rnodel. Suppose further that cornparison with our lists shows that the bifurcation diagrarns so generated are rnany or all ofthe qualitatively different perturbations of one specific qualitative type of equation, say g(x, A.) = 0. Then it is natural to conjecture that for sorne special values of the pararneters an equation equivalent to this g results. To verify such a conjecture one needs to solve a recognition problern, as was discussed above. If the conjecture is verified, then the physical pararneter space inherits useful structure frorn the universal unfolding, as was also discussed above. Typically this sequence of events leads to a cornpact description of a great deal of data. Following Thorn, we use the term organizing center to describe an equation type occurring in this way; i.e., an equation which occurs in a rnodel for certain values of the pararneters such that the universal unfolding of this equation generates rnany or all of the bifurcation diagrarns occurring in the physical problern. Each of the case sturlies illustrates the use of this concept in applications. We now outline the contents of this book, chapter by chapter. Chapters 11-IV, the essential theoretical core of the book, are a unit which develops the rnain ideas of the theory. These three chapters deal with the three problems discussed above; i.e., Chapters II, 111, and IV study the recognition problern, unfolding theory, and the classification problern, respectively. Chapter I highlights the theory to follow and discusses how singularity theory rnethods are used in applications. Also in this chapter we introduce the Liapunov-Schrnidt reduction in the lirnited context of ordinary dif-



ferential equations. (As we indicated above, with this technique many problems involving multiple solutions can be reduced to a single scalar equationg(x, .Ä.) = 0.) The style ofthe chapter is mainly expository, developing ideas by means of examples rather than theory. Chapter V explores a theoretical issue that singularity theory methods raise, the subject of moduli. Moduli are currently an active topic of research in several areas of pure rnathernatics. Regarding applications, rnoduli might at first seem to be an esoteric subject, but as illustrated by Case Studies 2 and 3, we have found rnoduli to play an irnportant role in the rnore interesting applications we have studied. (Remark: Chapter V considers rnoduli in the sirnplest context-one state variable with no symrnetry present. In applications, including Case Studies 2 and 3, moduli usually arise in a richer context involving syrnmetry.) Syrnrnetry and its consequences are the focus of Chapter VI. The restriction to one state variable greatly sirnplifies the discussion of syrnmetry since in one variable there is only one nontrivial symrnetry possible. Thus in this chapter we are able to illustrate, with a minimum of technical complications, the rnain issues involving syrnmetry. (The full complexities of syrnrnetry will be studied in Volurne II.) In particular, one point illustrated by Chapter VI is how singularity theory methods unify different contextsthis chapter uses the same methods as are used in the unsymmetric context of the preceding chapters, even though the specific results in Chapter VI are quite different from those of earlier chapters. Chapter VII develops the Liapunov-Schmidt reduction in general, expanding on the lirnited treatment in Chapter I. In this chapter we also illustrate the use oftbis reduction in applications-specifically, in a buckling problern and in certain reaction-diffusion equations. Chapter VIII studies Hopf bifurcation for systerns of ordinary differential equations; i.e., bifurcation of a periodic solution from a steady-state solution. This dynarnical problern can be formulated as a steady-state problem, thereby permitting the application of singularity theory methods. The advantageoftbis approachlies in the fact that these methods generalize easily to handle degenerate cases where one or more hypotheses of the classical Hopf theorem fail. Chapters IX and X together serve as a preview of the main issues to be studied in Volume II-bifurcation problerns in several state variables, especially with symmetry. The siruplest bifurcation problems in two state variables are discussed in Chapter IX, and certain bifurcation problems in two state variables with syrnrnetry are discussed in Chapter X. The treatment of these subjects is not cornplete; in particular, several proofs are deferred to Volume II. The three case studies in this book form an important part of it-they illustrate how singularity theory methods are used in applications. We believe that the three problerns analyzed in the case studies are of genuine scientific interest. (Other exarnples, of prirnarily pedagogical interest, have



been included within various chapters. Volume II will contain several more case studies, treating technically more difficult problems.) As to the interdependence of various parts, Chapters I-IV should be included in any serious effort to read the book. After this point there are some options. In particular, Chapters V, VI, and VIIare largely independent of one another, although the latter part of Chapter VI is closely related to Chapter V. By contrast, Chapter VIII depends heavily on Chapters VI and VII. Chapter IX may be read immediately following Chapter IV. (There is some reason to do so, as Chapter IX completes a theoretical development begun in Chapter IV; viz., the classification of bifurcation problems of codimension three or less. Chapter IX eliminates the restriction to one state variable that was imposed in Chapter IV.) Chapter X draws primarily on Chapter VI. Each case study is placed immediately following the last chapter on which it depends. In writing this book we wanted to make singularity theory methods available to applied scientists as well as to mathematicians-we have found these methods useful in studying applied bifurcation problems, especially those involving many parameters or symmetry, and we think others may too. Therefore we have tried to write the book in ways that would make it accessible to a wide audience. In particular, we have devoted much effort to explaining the underlying mathematics in relatively simple terms, and we have included many examples to illustrate important concepts and results. Several other features of the book also derive from our goal of increasing its readability. For example, each chapter and case study contains an introduction in which we summarize the issues to be addressed and the results to be derived. Likewise, in several places we have indicated material within a chapter that may be omitted without loss of continuity on a first reading, especially technically difficult material. In the same spirit, in cases where proofs are not central to the development, we have postponed these proofs, preferring first to discuss the theorems and give illustrations. Usually we have postponed proofs until the end of a section, occasionally until a later section, and in a few cases (the unfolding theorem among them) until Volume II. The prerequisites for reading this book may seem to work against our goal of reaching a wide audience. Regarding mathematical prerequisites, the text draws on linear algebra, advanced calculus, and elementary aspects of the theory of ODE, commutative algebra, differential topology, group theory, and functional analysis. Except for linear algebra and advanced calculus, we attempt to explain the relevant ideas in the text or in the appendices. Thus we believe it is possible for a nonmathematical reader to gain an appreciation of the essentials of the theory, including how to apply it, provided he or she is comfortable with linear algebra and advanced calculus. The many examples should help greatly in this task. Prerequisites for understanding the applications should not pose a problern for mathematical readers. Although our three case studies involve



models drawn from chemical engineering, mathematical biology, and mechanics, in each case we have described the physical origins of the equations of the model and then analyzed these equations as mathematical entities. A mathematical reader could follow the analysis of the equations without understanding their origins; of course some physical intuition would thereby be sacrificed. We are aware that many individuals whose work is not mentioned in this book have made important contributions to bifurcation theory. Consistent with our goals in writing this book, we have given references only when needed to support specific points in the text. Moreover by quoting one reference rather than another we do not mean to imply any historical precedent of one over the other-only that the quoted reference is one with which we are familiar and which establishes the point in question. The lack of a complete bibliography in this book is made less serious by the recent appearance of several monographs in bifurcation theory, for example, Carr [1981], Chow and Hale [1982], Guckenheimer and Holmes [1983], Hassard, Kazarinoff, and Wan [1981], Henry [1981], Iooss and Joseph [1981]. An amusing, personal anecdote may suggest further reasons why we have not attempted to include a complete bibliography. One of us was lecturing before an audience that included researchers in bifurcation theory. When asked to date a paper we had quoted, we guessed "around 1975." "lt was in the early sixties!" came the prompt reply from someone in the audience who had been associated with the work. Like children everywhere, we find that events before our time are somewhat blurred. There remains only the pleasant duty of thanking the many people who have contributed in one way or another to the preparation of this volume. Dave Sattinger originally suggested applying singularity theory methods to bifurcation problems. Jim Darnon has been a frequent consultant on the intricacies of singularity theory; moreover Lemma 2. 7 of Chapter III is due to him. Encouragement by, advice from, and lively discussion with John Guckenheimer, Jerry Marsden, and Ian Stewart have been most helpful. Joint work with our coauthors Barbara Keyfitz and Bill Langford are included in this text. The manuscript benefited greatly from suggestions made by Joe Fehribach and Ian Stewart. Barbara Keyfitz has contributed to the book in more ways than can reasonably be enumerated. To all these people, and to Giles Auchmuty, Vemuri Balakotaiah, Charlies Conley, Mike Crandall, Jack Haie, Phil Holmes, Ed Ihrig, Dan Luss, and Ed Reiss, we express our heartfelt thanks. The figures were drawn by Jim Villareal and Wendy Puntenney. While pursuing the research reported in this text, we were generously supported by the NSF and ARO, including visiting positions at the Courant Institute, the Institute for Advanced Study, the Mathematics Research Center, and the Universite de Nice. Finally we are grateful to Bonnie Farrell for her most efficient typing of an illegible manuscript-we only wish that we might have written the book as quickly, accurately, and cheerfully as she typed it.



A Brief Introduction to the Central ldeas of the Theory Introduction The Pitchfork Bifurcation The Continuous Flow Stirred Tank Reactor (CSTR) A First View of the Liapunov-Schmidt Reduction §4. Asymptotic Stability and the Liapunov-Schmidt Reduction

§0. §I. §2. §3.



16 25 35


The Implicit Function Theorem



Equivalence and the Liapunov-Schmidt Reduction



The Recognition Problem §0. §1. §2. §3. §4. §5. §6. §7. §8. §9. §10. §11. §12. §13.

Introduction Germs : A Preliminary Issue The Restricted Tangent Space Calculation of RT(g), I: Simple Examples Principles for Calculating RT(g), I: Basic Algebra Principles for Calculating RT(g), li: Finite Determinacy Calculation of RT(g), li: A Hard Example Principles for Calculating RT(g), III: Intrinsic Ideals Formulation of the Main Results Solution ofthe Recognition Problem for Several Examples The Recognition Problem: General Equivalences Proof of Theorem 2.2 Proof of Proposition 8.6(b) Proof of Theorem 8.7

51 51 54 56 58 63 66 77

81 86 93 96 98 102 107




Unfolding Theory §0. §1. §2. §3. §4. §5. §6. §7. §8. §9. §10. §11. §12.

Introduction Unfoldings and Universal Unfoldings A Characterization of Universal Unfoldings Examples ofUniversal Unfoldings The Recognition Problem for Universal Unfoldings Nonpersistent Bifurcation Diagrams Statement of the Main Geometrie Theorem A Simple Example: The Pitchfork A Complicated Example: The Winged Cusp A Sketch of the Proof of Theorem 6.1 Persistence on a Bounded Domain in a Parametrized Family The Proof of Theorem 10.1 The Path Formulation

117 117

120 123 129


140 143



152 154

159 165





Classification by Codimension §0. §1. §2. §3.

Introduction Philosophical Remarks Concerning Codimension The Classification Theorem Universal Unfoldings of the Elementary Bifurcations §4. Transition Varieties and Persistent Diagrams

182 182 183 196 202



An Example of Moduli §0. §I. §2. §3. §4. §5. §6. §7.

Introduction The Problem of Moduli: Smooth Versus Topological Equivalence The Recognition Problem for Nondegenerate Cubics Universal Unfolding; Relation to Moduli Persistent Perturbed Diagrams A Picture of the Moduli Family Discussion of Moduli and Topological Codimension The Thermal-Chainbranching Model

213 213 214 216 219

222 226 236 239


Bifurcation with Z 2 -Symmetry §0. §I. §2. §3. §4.

§5. §6. §7. §8.

Introduction A Simple Physical Example with Z 2 -Symmetry The Recognition Problem Universal Unfoldings Persistent Perturbations The Z 2 -Classification Theorem Persistent Perturbations of the Nonmodal Bifurcations The Unimodal Family ofCodimension Three Perturbations at the Connector Points

243 243 244 247 257

260 263





The Liapunov-Schmidt Reduction §0. Introduction

§1. §2. §3. §4.

The Liapunov-Schmidt Reduction Without Symmetry The Elastica: An Example in Infinite Dimensions The Liapunov-Schmidt Reduction with Symmetry The Liapunov-Schmidt Reduction of Scalar Reaction-Diffusion Equations §5. The Brusselator §6. The Liapunov-Schmidt Reduction of the Brusselator


289 289 290 296 300 308 312 317


Srfiooth Mappings Between Banach Spaces



Some Properties of Linear Elliptic Differential Operators



The Hopf Bifurcation §0. lntroduction

§I. §2. §3. §4. §5.

Simple Examples of Hopf Bifurcation Finding Periodic Solutions by a Liapunov-Schmidt Reduction Existence and Uniqueness of Solutions Exchange of Stability Degenerate Hopf Bifurcation

337 337 338 341 350 358 372


The Clamped Hodgkin-Huxley Equations



Two Degrees of Freedom Without Symmetry §0. Introduction

§I. Bifurcation with n State Variables §2. Hilitop Bifurcation §3. Persistent Bifurcation Diagrams for Hilitop Bifurcation

397 397 398 400 407


Two Degrees of Freedom with (Z 2 E9 Z 2 )-Symmetry §0. Introduction

§1. §2. §3. §4.

Bifurcation Problems with (Z 2 Ef) Z 2 )-Symmetry Singularity Theory Results Linearized Stability and (Z 2 Ef) Z 2 )-Symmetry The Bifurcation Diagrams for Nondegenerate Problems

417 417 418 421 426 430


Mode Jumping in the Buckling of a Reetangular Plate




Symbol Index