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Chaotic Dynamics: An Introduction Based on Classical Mechanics

Tamás Tél and Márton Gruiz
Cambridge University Press
Publication Date: 
Number of Pages: 
[Reviewed by
David S. Mazel
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This book is an outstanding text describing and detailing chaotic systems from a classical mechanics viewpoint. It is clear, well-organized, well-written, and filled with examples, pictures, graphs, and problems for different chaotic systems. Whether the reader just reads through it or, better still, if he/she works through the examples, this book will give the reader a broad background on chaotic systems and provide analytical skills and insights necessary for independent study or research.

The list of chapters can be seen in the table of contents . Each chapter is filled with explanatory text, figures that are extremely well-done and illustrative (more so than I have seen in other similar books), and problems that take the concepts further than the text. At the back of the book there are solutions to most, but not all, of the problems.

Here is why I found this book so good.

The introductory chapter shows how chaos is prevalent in nature, giving simple physical examples of it. Examples include the driven oscillator, a driven pendulum, and wedge billiards. Each example is easy to understand, particularly with the illustrations, and opens the door to the required mathematics for analysis. The authors describe phase space and fractals in sufficient detail for beginners to understand and they do so without diverting the reader's attention from the chaotic systems that the book is really about.

Concepts are well-defined and illustrated with pictures. For example, features of phase space such are hyperbolic points are shown with small pictures alongside the text that beautifully combine math, picture, and text in a single paragraph without breaks. (This style, by the way, is reminiscent of Edward Tufte's desire for better presentation of information; see here for a review of his latest book.) Along with the clear explanations are "boxes" of special, often tangential discussions, of related ideas that are important yet not directly related to the flow of the text.

For example, this is how the authors discuss the water-wheel. To begin, the water wheel is a heavy circular wheel with buckets distributed along the rim — this is shown with a drawing. A steady rain falls so that the buckets fill with water but the buckets have a flaw — each has a hole in its bottom so that water leaks out even as the bucket fills. The question is: how does the wheel move for a given rainfall and leak-rate? To answer the question, the authors derive the nonlinear equations of motion in explicit detail. Next, the authors find the fixed points of motion and explore the behavior for different rain fall intensities. Finally, the chaotic nature of the wheel is detailed in equations and illustrated with seven different figures. Other systems are similarly well-analyzed and described.

(I must give the authors credit for doing what I have wanted to see for a while: a clear introduction to billiards of various types such as planar tables, and scatters, with associated terminology clearly defined.)

The authors conclude with applications of chaos such as the three body problem for spacecraft (a review of a book on this topic is here) and vortex dynamics. At end of the book, the authors even provide programming examples, albeit in Pascal which is not often used today, that can help the reader to code the examples if he/she chooses to do so. Selected solutions to the problems finish the book.

Additional thoughts: As an introduction, this book is excellent. It is clear, detailed, and well-illustrated. The systems are easy to visualize and understand even though their behavior is anything but simple. The mathematics are readily accessible to an undergraduate (with some effort but not too much as to be prohibitive) and the figures are plentiful, well done, and clearly illustrate the concepts.

Anyone beginning his/her study of chaotic systems would do well to have read this book or at least have it as a reference. There are many books today that profess to be an introduction, this one truly is and is even more.

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.

 Introduction; Part I. The Phenomenon: Complex Motion, Unusual Geometry: 1. Chaotic motions; 2. Fractal objects; Part II. Preparatory Concepts: 3. Regular motions; 4. Driven motions; Part III. Investigation of Chaotic Motion: 5. Chaos in dissipative systems; 6. Transient chaos in dissipative systems; 7. Chaos in conservative systems; 8. Chaotic scattering; 9. Applications of chaos; 10. Epilogue, outlook; Part IV. Miscellaneous: 11. Appendices; 12. Solutions to problems; 13. Bibliography.