book     Book Review

Chaos: An Introduction to Dynamical Systems


Kathleen T. Alligood, Tim D. Sauer and James A. Yorke
Springer-Verlag, New York, 1997.
603 pp. $39. 00 hc
ISBN 0-387-94677-2

Kathleen Alligood, Tim Sauer and James Yorke's Chaos is a well-written book that provides a detailed introduction to dynamical systems theory, with a strong emphasis on dissipative systems and low-dimensional chaotic dynamics. The authors are mathematicians and well known to the nonlinear dynamics community; in particular, Yorke has for more than 20 years played a seminal role in the development of chaos theory.

As might be expected from such a trio, the book presents the mathematical theory with care. The presentation nonetheless retains an introductory flavor throughout, by relying on examples to convey the main ideas. Theorems are stated and often proved, but in a reasonably humane manner: No effort is made to formulate results in their most general setting, and the proofs are carefully explained. On some points, such as the lambda lemma or the differentiability of the stable manifold, where any proof is necessarily rather technical, the authors provide references in which the arguments can be found.

Although the widespread interest in nonlinear dynamics has produced a number of books on the subject, very few offer this level of mathematical detail and rigor packaged within a genuinely introductory discussion. The organization of the material has several notable characteristics: Overall, the subject is presented in what I would call reverse order: The opening chapters treat chaos in maps, fractal sets and various bifurcations (such as crises) that arise in the examples; in later chapters, we find flows in one- and two-dimensional phase spaces and a description of the elementary bifurcations from fixed points. I would guess that this order might be hard on some students, but I have not tried it in my own teaching. It is undeniably economical, if you want to get to the sexy topics fast.

While their main focus is on the mathematics and the study of various model equations, the authors convey the relevance to real experiments through 12 "lab visits" that appear as chapter appendices. These are brief sketches of the appearance of the dynamical phenomena when seen in the flesh, with examples chosen from chemistry, physics and biology.

A second set of chapter appendices, written in the form of extended homework exercises, expand on the theoretical development of each chapter. These "challenges" include the proof of Sharkovskii's theorem, the application of shadowing to justify numerical computation of chaotic trajectories and the analysis of synchronization between coupled chaotic systems.

In addition to covering standard topics, also found in many other texts, there are quite readable introductions to a number of advanced topics. In the context of well-chosen examples, the authors describe Markov partitions, invariant measures and natural measure, shadowing, periodic orbit cascades and so-called Wada basins (whose fractal boundaries have a structure that is almost beyond belief). Many of these examples derive from the research of the authors.

The book concludes with a chapter on phase-portrait reconstruction from experimental time series and a discussion of various embedding theorems. The book omits some things that I like to include in an introductory graduate course. Surprisingly, the concept of structural stability is not developed. As a result, although the pages are filled with examples of bifurcations, the concept of bifurcation as a change in the topological equivalence class is never formulated. Similarly, Peixoto's theorem on two-dimensional flows is not stated. The discussion of elementary bifurcations for maps and vector fields omits center manifolds and normal forms entirely, so the reader cannot appreciate the full significance of the simple one- and two-dimensional examples. Hopf bifurcation in maps is not presented. The importance of homoclinic points is stressed, but Melnikov methods for detecting them are not mentioned. In a discussion of a damped driven pendulum, the main point of which is to introduce a return map with fractal basin boundaries, the authors miss an opportunity to explain how dimensional analysis allows a model to be simplified. Instead we are told that to obtain the dimensionless model, it is necessary to "use a pendulum of length l = g." How to achieve this feat is presumably an exercise for the reader!

I think the text should be quite useful for physics graduate courses and honors-level undergraduate courses, although it contains far more material than could be covered in a single term. By the same token, it is a serious introduction, with wider coverage of the subject than is readily available from any other single source.

University of Pittsburgh
Pittsburgh, Pennsylvania

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