UK Nonlinear News, February 2000

Dynamical Systems and Numerical Analysis

By A.M. Stuart and A.R. Humphries

Reviewed by Jaroslav Stark

Cambridge University Press 1999, ISBN 0-521-664563-8, 685 pages, #24.95.
Previously published in hardback: 1996, ISBN 0-521-49672-1, #45.00

The two main factors underpinning the dramatic growth of nonlinear dynamics in the second half of this century have undoubtedly been on one hand the introduction of qualitative, and in particular topological, methods and on the other the increasing power and availability of digital computers. Surprisingly, although there is a wide range of books covering the former, there has been an acute shortage of texts dealing with the numerical aspects of nonlinear dynamics. This includes a lack of both practical textbooks explaining how to use computers to help understand the behaviour of nonlinear systems, and of more theoretical publications discussing to what extent numerical simulations accurately represent the behaviour of a dynamical system. The latter is particularly serious: most traditional numerical analysis texts are primarily concerned with accuracy over a fixed time period, and hence are of only marginal interest in nonlinear dynamics where one is concerned with long time asymptotic behaviour. Conversely, most books on dynamical systems largely ignore the issue of whether what one sees on a computer is a genuine phenomenon, or a computer artefact.

As far as I am aware, the research monograph under review is therefore the first attempt to provide a comprehensive account of this topic. It is written by two of the leading experts in the field, who are responsible for many of the results it contains. The appearance of such a work is thus very welcome and long overdue, and will undoubtedly help to stimulate a wider appreciation of the problems involved in the numerical integration of differential equations. The publication of a paperback edition will also help to make this volume more accessible to research students and hence hopefully motivate a new generation of mathematicians to work in this important field.

Despite containing a vast amount of material and detailed technical derivations, the volume is well written and interesting to read. From a personal perspective, I would have benefited from more motivation for the various different integration schemes and types of numerical stability discussed by the authors. However, this perhaps reflects my relative ignorance of numerical analysis. A more serious lack is the omission of any kind of conclusions, or of a relative comparison between the merits of different numerical methods: anyone coming to this book from nonlinear dynamics looking for advice for which integration scheme to use will be disappointed. I also found the lack of discussion of structural stability rather strange, given the fundamental role that this notion has played in the development of modern dynamical systems and its close relationship to many of the ideas in the book. Similarly, shadowing receives only a very brief mention, despite representing a central concept in modern dynamics and increasingly providing a standard framework for the investigation of precisely the problems raised in this book. Having said this, the volume is already nearly 700 pages long, and hence the authors had to stop somewhere. Introducing further material would probably have detracted from the coherent flow of ideas that they have managed to achieve. Despite these minor caveats I am therefore happy to recommend this volume to all those interested in the relationship between dynamical systems and computation. In particular, it ought to be required background reading for all new research students who will be using numerical methods to investigate the behaviour of nonlinear dynamical systems.

Jaroslav Stark,
Monday, November 1, 1999.

UK Nonlinear News thanks Cambridge University Press for providing a copy of this volume for review.

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Last Updated: 6th February 2000.