UK Nonlinear News, August 1999

Elements of Applied Bifurcation Theory (Second Edition)

By Yuri A. Kuznetsov

Reviewed by Harvinder Sidhu

Published by Springer-Verlag
Applied Mathematical Sciences 112
ISBN: 0-387-98382-1

According to the UK Nonlinear News list of nonlinear book reviews, the first edition of this book has been reviewed by Alan Champneys (in UK Nonlinear News), David Arrowsmith (in Bulletin (New Series) of the American Mathematical Society 33(3):377-380, 1996) and Eusebius Doedel (in Nonlinear Science). Furthermore, Alan Champneys has already provided UK Nonlinear News with his views regarding to the second edition. After careful deliberation I have decided to focus my review strongly on the second edition, with occasional reference to the first (particularly when there have been significant additions)

The book possesses 10 chapters dealing with bifurcation theory and its application to discrete-time and continuous-time dynamica systems. It does this at both an elementary and an advance level. This is precisely what makes this book so appealing to both researchers and students with varied mathematics backgrounds. The examples throughout the book are generally well chosen to illustrate complex concepts. I have found this book to be an invaluable addition to my library, referring to it in both my research and in my teaching. The bibliographical notes at the end of each chapter are very informative, providing the reader with not only historical background but also results from recent work.

The book begins with a chapter dedicated to introducing the reader to dynamical systems. Basic concepts such as orbits and invariant sets are introduced without reference to differential equations. Chapter 2 introduces and discusses the notion of topological equivalence and their classification. The concept of bifurcation and bifurcation diagrams are also introduced. This chapter concludes with discussions relating to topological normal form and structural stability. The basics concepts of chapter 1 and chapter 2 are exceptionally well introduced.

With the introductory part of the book out of the way the rest of the book deals with bifurcations in dynamical systems. Chapter 3 examines the simplest types of bifurcation of equilibria in a continuous-time system: the fold and Hopf bifurcations. The proof of the topological equivalence of the original and truncated normal forms for the fold bifurcation was absent in the first edition. This has now been included, together with an example of the Hopf bifurcation analysis in a planar system using MAPLE.

Chapter 4 is very much like the previous chapter except that it deals with maps and the simplest possible bifurcations of fixed points, ie. the fold, flip and Neimark-Sacker bifurcations. The detailed normal form analysis of the Neimark-Sacker bifurcation is illustrated in the delayed logistic map.

The main focus of chapter 5 is to show how the bifurcation of a dynamical system of higher dimension can be reduced to the system's center manifold. Although the proof of the center manifold theorem is absent, the examples and discussions illustrate the basic concepts exceptionally well. The explicit formulae for the normal form coefficient of all codimension 1 bifurcations of n-dimensional maps are presented.

The idea of global bifurcations corresponding to homoclinic and heteroclinic orbits is presented in chapter 6. Here the two- and three-dimensional cases are discussed first to allow the reader to gain some geometrical intuition, before proceeding to higher dimensions. The section on homoclinic bifurcations in n-dimensional ODEs has been completely re-written so as to introduce the Melnikov integral, which allows the verification of the regularity of the manifold splitting under parameter variations. Recently proved results on the existence of center manifolds near homoclinic bifurcations are included.

Chapter 7 covers several distinct topics relating to one-parameter bifurcations in continuous-time systems. These include: global bifurcations of orbits that are homoclinic to nonhyperbolic equilibria; bifurcations on invariant tori; phase-locking and Arnold tongues. There is some discussion of bifurcations of equilibria and cycles in the presence of the simplest symmetry group, Z2. I particularly liked the example illustrating the "blue sky" bifurcation.

The treatment of two-parameter bifurcations of both continuous and discrete systems are discussed in chapters 8 and 9 respectively. These two chapters alone constitute about 1/3 of the book. The cusp, Bogdanov-Takens, Bautin, fold-Hopf, and Hopf-Hopf bifurcations are all analysed in chapter 8. Although these are complex concepts, Kuznetsov's utilisation of figures showing the phase-portraits of these bifurcations makes these ideas more accessible to the reader. Chapter 9 dealt with cusp, flip and Neimark-Sacker bifurcations in maps.

The final chapter is devoted to the numerical methods commonly used in bifurcation analysis of dynamical systems. Here the focus is mainly on continuous-time systems. Numerical techniques discussed include the location and stability analysis of equilibria, and limit cycles. Several techniques for equilibrium continuation and detection of codimension 1 bifurcations, including continuation of homoclinic orbits are also presented. Furthermore the second edition introduces boundary-value methods to continue codimension 1 homoclinic bifurcations in 2 parameters and codimension 1 limit cycle bifurcations. A new appendix has been added to provide test functions to detect codimension 2 homoclinic bifurcations involving a single homoclinic orbit to an equilibrium. The software review which appears in Appendix 3 has also been updated.

This is an excellent book for researchers and students who are interested in dynamical systems and bifurcation theory, and I would not hesitate to recommend it to anyone who is interested in this area.

UK Nonlinear News thanks Springer-Verlag for providing a copy of this book for review.

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Last Updated: 30th July 1999.