Department of Applied Mathematics University of Leeds, Leeds ls2 9jt, UK

UK Nonlinear News Book Review

Chaotic Dynamics: An Introduction Based on Classical Mechanics

Tamás Tél and Márton Gruiz

Recalling my undergraduate mechanics lectures I half-expected a book full of complicated trigonometry, lengthy equations, peculiar coordinate systems and weighty analysis. What I read, and enjoyed, was a well-organised, beautifully illustrated volume covering the main points of chaotic dynamical systems, using intuitive mechanical examples to motivate clear and lucid discussions.

The target audience of the book is undergraduate students of science, engineering and computational science, and one of its stated aims is to ''contribute to clarifying some misconceptions arising from everyday usage of the term 'chaos'.'' This it fulfills handsomely. It is certainly a book intended to teach or learn from, rather than as a research reference text. Indeed its opening chapter even contains a list of instructions on how to examine chaotic motion. Despite this, the book does far more than simply ``go through the motions''. Diagrams are plentiful and detailed, which is crucial for the geometric viewpoint presented. Many well-known mechanical systems (for example, the magnetic pendulum, the water-wheel, the spinning top) introduce concepts which are first abstracted and then used to discuss, albeit briefly, a wider range of applications, including environmental issues, chaotic scattering and the three-body problem. Throughout the work self-contained text capsules appear, expanding on issues raised in the discussions, and containing excellent summaries, histories, explanations and more potential applications.

One of the book's main strengths is its excellent layout and ordering. It is unusual but pleasing to see the book open with a chapter describing the temporal phenomenon of chaotic motion, followed by a new and entirely separate chapter discussing the geometric phenomenon of fractal structures. Too often are the terms ''chaotic attractor'' and ''strange attractor'' mistakenly taken to be synonyms. The student reading this book is left with a thorough understanding of the connections and differences between the two concepts. At this stage the reader has been armed with the basic ideas of chaotic motion and fractal geometry, but is not yet permitted to attack problems displaying these traits. First, the authors introduce the necessary ideas from regular and periodically driven motion that will be needed, stressing the importance of studying unstable behaviour. All of this background material is presented at a level suitable for undergraduates.

Inevitably there are omissions. Despite the chapter on driven motion there is no mention, for example, of synchronization, which is perhaps surprising given in the mechanical setting. The emphasis is on firmly presenting tools used to study chaotic systems, rather than giving a rigorous mathematical treatment. For example, while stable and unstable manifolds are discussed in terms of their use in dividing phase space, the reader must look elsewhere for results on the existence and smoothness of such objects.

I would happily lend this book to any student wishing to begin learning the subject of chaotic dynamics, but I would certainly also demand its return.

Reviewed by Rob Sturman
Department of Applied Mathematics
University of Leeds.

UK Nonlinear News would like to thank Contemporary Physics for allowing a copy of this review to appear here.