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Concepts and Results in Chaotic Dynamics: A Short Course

Pierre Collet and Jean-Pierre Eckmann
Springer Verlag
Publication Date: 
Number of Pages: 
Theoretical and Mathematical Physics
[Reviewed by
David S. Mazel
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Two thoughts crossed my mind when I picked up this book. The first was: "what a physically attractive book." The second was: "what a short book to have on such a wide ranging topic."

Physically, and this is often not discussed in a review, the book is wonderful. It is a perfect size to carry in a knapsack, the print is clear and the layout of text, equations, and figures is marvelously done. Some of the figures, for example figure 2.1 of the flow for a nonlinear pendulum, "give a 3-dimensional effect if one 'stares' at them to bring images of the two cylinders to convergence." These images are multi-colored stereo images, and allow the reader to "see" a three dimensional effect that helps illustrate the phenomena. The authors make good use of color to show mixing effects, basins of attraction, and contractions. These images help to understand the text. Other uses of color, in later figures, are good but not entirely necessary. As a physical book, this one is especially pleasant and easy to read. Its shortcomings are in terms of content.

The book is short, only 214 pages of text. It covers such a wide range of topics, however, that the discussions are often terse and the notation can be confusing. The first four chapters (A Basic Problem, Dynamical Systems, Topological Properties, and Hyperbolicity) are satisfying to the knowledgeable reader but lack details that would make them accessible to an undergraduate or beginning graduate student. I think beginning researchers would find these discussions less than appealing as well.

The final five chapters (Invariant Measures, Entropy, Statistics and Statistical Mechanics, Other Probabilistic Results, and Experimental Aspects) are difficult to follow, with confusing notation and limited background discussion. This approach may be in keeping with the subtitle, "A Short Course," but more explanatory text would have been a plus. For example, the last chapter mentions the power spectrum and the periodogram, which the authors quickly define. But they do not discuss any of the properties or utility of these techniques. Consequently the reader is left to wonder whether these ideas are of practical use or not.

In brief, the book covers a wide range of ideas. However, it would have been better for the authors to explore these ideas in more detail by either expanding the entire text or breaking the book into two separate books, say, a course I and a course II with added details and examples in each.

David S. Mazel received his doctorate from the Georgia Institute of Technology in electrical engineering and is a practicing engineer in Washington, DC. His research interests are in the dynamics of billiards, signal processing, and cellular automata.