UK Nonlinear News, August 1999

Introduction to Chaos

By Hiroyuki Nagashima and Yoshikazu Baba

Reviewed by Cliburn Chan

Institute of Physics Publishing, 1999
Translated and revised from the Japanese edition
Baifukan Co., Ltd, 1992
Hardback: ISBN 0 7503 0507 X
Softback: ISBN 0 7503 0508 8

This slim book subtitled ‘Physics and Mathematics of Chaotic Phenomena’ has a stated target audience of advanced undergraduate students in science, and was written as a collaboration by a mathematician and a physicist. The fact that this book has only 4 chapters but 17 appendices suggests that it would take a brave undergraduate to plough through it alone.

Although only 168 pages in length, the book covers a lot of ground extremely rapidly. Chapter 1 has only 12 pages and gives a general introduction, but ends with the following ominous characterisation of chaos:

  1. A condition given by Li and Yorke is ‘if a map has periodic motions with the period 3, it leads to chaos’,
  2. Related to (1) is the ‘positivity of the ‘topological entropy’ and
  3. ‘positivity of the Lyapunov exponent given by the logarithm of the expansion rate of the map’

Chapter 2 is the only one written by the mathematician, and is noticeably harder going than the rest of the book. By the end of chapter 2 (page 41), the intrepid reader has been taken through the Li-Yorke theorem, Sharkovski’s theorem, invariant measures and topological entropy. The price for this is that the uninitiated reader (i.e. me) is constantly referring to the appendices for the necessary background to follow the discussion.

Chapter 3 gives an account of the Feigenbaum and intermittent routes to chaos using the logistic map. The discussion is well-paced and clear, and the copious illustrations have been well chosen to aid understanding.

The final chapter extends the discussion of chaos to flows, using the simple harmonic oscillator to show the relationship between phase volume and conservative or dissipative systems. Strange attractors and the use of Poincaré sections are illustrated with Rössler’s equations. The book ends with a description of the tools of nonlinear time series analysis, including state space reconstruction by embedding, dimensional analysis and calculation of the Lyapunov spectrum.

The line drawings and monotone computer graphics are extremely attractive, and contribute immensely to understanding. Occasional Mathematica programs allow motivated readers to generate similar graphics for themselves. The problem sets are few but interesting, and solutions are provided to pacify the perplexed.

Overall, I quite enjoyed this little volume, and think it provides a sound mathematical foundation for further excursions into the world of nonlinear dynamics. As such, it nicely complements the more application-oriented undergraduate textbooks like Nonlinear Dynamics and Chaos by Strogatz, and Understanding Nonlinear Dynamics by Kaplan and Glass.

Cliburn Chan
Centre for Nonlinear Dynamics and its Applications
University College London

UK Nonlinear News thanks the Institute of Physics for providing a copy of this book for review.

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Last Updated: 2nd June 1999.