Book Review

Source: Zentralblatt fur Mathematik, 1997

Chaos. (English)
An introduction to dynamical systems.

Springer-Verlag, New York, 1997. xviii+603 pp. DM 64.00; $39.00. ISBN 0-387-94677-2

By Kathleen T. Alligood, Tim D. Sauer and James A. Yorke.

This is a well-presented introductory text for undergraduates/beginning graduates: it succeeds to avoid the standard examples of such texts, and, instead, presents nonlinear dynamics from a different viewpoint. Roughly the first half of the book deals with maps in one and two dimensions, and discusses bifurcation and chaotic behavior, fractals and attractors. The second part discusses similar concepts for ordinary differential equations, and introduces topics such as periodic orbits and limit sets, different kinds of bifurcations, chaos, stable manifolds and crises. Additional chapters deal with bifurcation cascades in maps and experiments and the state reconstruction from experimental or numerical data. The book can be recommended as an introductory text for a number of reasons: -- it presents the material thoroughly and comprehensively; -- it contains many new and unusual (solved) exercises; -- the ``lab visits'' at the end of each chapter familiarize the reader with a number of recent research results; -- the suggested computer experiments are well-explained so that they can be carried out easily on a personal computer; -- the explanations of new terms (``What is a manifold?'') in separate boxes are an invaluable help to the beginner.