Oxford University Press, 2003, ISBN 0-19-850839-5, xx + 507 pp., $120.00.

The theme of this book is the detection and characterization of chaos in dynamical systems, based on measurements of the system state as a function of time. It would be appropriate for a physics course for either advanced undergraduates or graduate students. With supplementary material, perhaps from Sprott (1993), the same author, out of print but available at https://sprott.physics.wisc.edu/sa.htm, it could form the basis of a course in applied mathematics or mathematical physics.

This is a very ambitious book, covering a huge amount of material. The first 8 chapters are a comprehensive introduction to dynamical systems, emphasizing concepts and practical matters with the necessary mathematics being provided as needed. Chapter 9 begins the treatment of time series, followed by clear and detailed discussions of nonlinear prediction and noise reduction. After an introduction to fractals, there are chapters on the computation of fractal dimension, fractal measures and multifractals, and nonchaotic fractal sets.

For certain research interests, the treatment of spatiotemporal chaos and complexity in the final chapter is rather superficial. This brief discussion does cover the generation of complexity and organization in the evolution of sample spatially extended systems, but does not do justice to the theme of the book: how such systems can be studied with time series analysis -- e.g. using symbolic dynamics and related time series approaches, such as developed by Crutchfield and his co-workers (Crutchfield and MCNamara 1987, Young and Crutchfield 1993).

Finally, I believe that more warnings should be given to the person interested in the actual analysis of time series data. Especially in astronomy, many researchers went wrong by naively assuming that estimation of smallish, non-integer fractal dimensions implied the presence of chaotic dynamics. The papers by Osborne and Provenzale in the book's bibliography, and one by Eckmann and Ruelle (1993), point out the pitfalls that face the analyst of finite, noisy data.

But these are very minor objections. On the whole, this is a masterful volume that will be very useful for students at various levels, as well as for researchers. I believe that this is the first book to systematically cover analysis of time series data from chaotic dynamical systems, and is therefore a very welcome publication indeed.

Jeffrey D. Scargle

Space Science Division

NASA-Ames Research Center

REFERENCES

Crutchfield, J. P., and McNamara, B. (1987),

"Equations of Motion from a Data Series", Complex Systems, 1, pp. 417-452.

Eckmann, J.-P., and Ruelle, D. (1992),

"Fundamental limitations for estimating dimensions and

Lyapunov exponents in dynamical systems,"

Physica D, 56, pp. 185-187.

Sprott, J. C. (1993), Strange Attractors: Creating

Patterns in Chaos, New York: M&T Books.

Young, K. and Crutchfield, J. P. (1993), "Fluctuation Spectroscopy",

Chaos, Solitons, and Fractals, 4, pp. 5-39.

Jeffrey.D.Scargle@nasa.gov