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Elements of Applied Bifurcation Theory, Second Edition Yuri A. Kuznetsov Springer To my family This page intentionally left blank 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.Thisneweditionpreservesthestructureofthefirsteditionwhile 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 ele- mentary 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. ThischapteralsoincludesanexampleoftheHopfbifurcationanalysisina 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 for- mulas 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 inte- gral 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 analysisofbifurcationsinfour-dimensionalsystemswithahomoclinicorbit 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 thenumericalanalysisofbifurcations,hasbeenchangedmostsubstantially. 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 gradi- ent 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 testfunctionstodetectallcodim2homoclinicbifurcationsinvolvingasin- gle 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. Anumberofmisprintsandminorerrorshavebeencorrectedwhileprepar- ing 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, En- schede), B. Sandstede (WIAS, Berlin), W.-J. Beyn (Bielefeld University), F.S.Berezovskaya(CenterforEcologicalProblemsandForestProductivity, Moscow),E.NikolaevandE.E.Shnoll(IMPB,Pushchino,MoscowRegion), 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 work- ing on this book and developing the bifurcation software CONTENT. Finally, I would like to acknowledge the Research Institute for Applica- tions 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 ori- entedandleavemanypracticalissuesunclearforpeopleintendingtoapply the theory to particular research problems. This book is designed for ad- vanced undergraduate or graduate students in mathematics who will par- ticipate in applied research. It is also addressed to professional researchers inphysics,biology,engineering,andeconomicswhousedynamicalsystems asmodelingtoolsintheirstudies.Therefore,onlyamoderatemathematical backgroundingeometry,linearalgebra,analysis,anddifferentialequations 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 thebook.Wheneverpossible,onlyelementarymathematicaltoolsareused. For example, we do not try to present normal form theory in full general- ity, 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 cov- ered 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-