Chaos and Time-Series Analysis, by Julien Clinton Sprott,
Oxford University Press, 2003, ISBN 0-19-850839-5, xx + 507 pp.,
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 http://sprott.physics.wisc.edu/sa.htm,
it could form the basis of a course in applied mathematics or
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
Crutchfield, J. P., and McNamara, B. (1987),
"Equations of Motion from a Data Series", Complex Systems, 1, pp.
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.