UK Nonlinear News, May 1999

Some comments on Elements of Applied Bifurcation Theory (2nd edition)

By Yuri A Kuznetsov

Reviewed by A.R. Champneys

Springer-Verlag, 1998.
Applied Mathematical Sciences 112
Hardback: ISBN 0-387-98382-1

I wrote a (not exactly impartial) review for the first edition of this book for an earlier issue of UK Nonlinear News. Since then I have found the book to be an invaluable tool both in my own research and also to recommend to students (both as a general graduate level introduction to dynamical systems and as a practical manual for calculation on example systems). Nevertheless, reading the review I wrote then reminded me of the euphoria I feel after just finishing a great novel or listening to a new piece of music. A while afterwards you are liable to feel embarrassed at the unquestioning praise you lavished.

So, since UK Nonlinear News asked me, I have recently attempted to look the second edition with fresh eyes (forgetting that I have used the first edition so much and forgetting that Yuri is my friend). I must admit that I still believe this is a great book.

It is comprehensive in its treatment of local bifurcations in generic finite dimensional systems (it is the only place I am aware of that gives the reader all the necessary information for performing centre manifold and normal form reductions for themselves). The illustrations are superb, as they need to be in what is after all a geometric subject, and there is a strong emphasis on numerical methods. It is both a toolkit and a primer.

The alterations between first and second editions are fairly minor. Various explicit formulae for the calculation of normal form coefficients are given for the first time. Also, global bifurcations in Chapters 6 and 7 get something of a re-working. In keeping with the comprehensive treatment of local bifurcations, homoclinic bifurcations are now treated more systematically by first looking at all cases in 2,3 and 4 dimensions before proceeding to arbitrary dimensions by use of a the Melnikov integral and a recently proved `homoclinic centre manifold' theorem. There is a wonderful new example of a blue sky catastrophe. Chapter 10, on numerical methods has been expanded in several ways incorporating some of the latest research and methods on linear algebra for bifurcation continuation in 2 or more parameters, and on boundary-value methods for limit cycle and homoclinic bifurcations. Many of these techniques are included in Yuri's bifurcation software package CONTENT which I would also recommend (the latest version CONTENT 1.5 is freely available by anonymous ftp from ( in the directory pub/CONTENT. See the README file there).

I would envisage that this book will continue to be of use to researchers and students in dynamical systems for many years to come. Who knows, maybe by the time of the n^{th} edition, UK Nonlinear News will stop asking me to review it. I hope not.

Alan Champneys
Dept of Engineering Maths
University of Bristol
Bristol BS8 1TR UK

UK Nonlinear News thanks Springer-Verlag for providing a copy of this book for review (to be reviewed next issue).

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