Book Review: The theory of chaotic attractors

Abstract

James Yorke won the 2003 Japan Prize for his work in the field of chaos theory. This book was compiled by four of his best-known collaborators in honor of his 60th birthday and contains papers by various authors on chaos and chaotic attractors. The papers are organized around the topics of "natural" invariant measure and fractal dimension of the associated chaotic attractors. A few of the papers are written by Yorke and collaborators, but the editors say they have chosen the collection primarily for the papers' historical importance and their accessibility to students.The book begins with the famous paper "Deterministic Nonperiodic Flow" by Edward Lorenz. Anyone who doubts the importance of interdisciplinary institutes or of "linking" figures like Yorke should read Gleick's [4] account of Yorke's role in the dissemination of this paper to the mathematics community. The shocking aspect of the story is that the paper had been in print a full 9 years before most mathematicians were aware of it. Lorenz had found a three-dimensional ODE (a reduction of a model for Rayleigh-Benard instability in hydrodynamics) containing a limit set A that appeared to have the following properties:(1) The orbit of almost every initial condition in a neighborhood of A has its limit points in A (hence, it is an attractor).(2) The limit set is not periodic or asymptotically periodic.(3) The orbits show sensitive dependence on initial conditions. (Hence, by (2) and (3), the limit set A is chaotic.) (4) The limit set A contains a dense orbit (hence, it is indecomposable). Although several properties of a chaotic attractor are listed here, there is no universally accepted definition. Some prefer a more mathematical (technical) statement; others want conditions that could be verified by experimentalists. In a paper which appears in this volume, Milnor gives a definition of "attractor" in which condition (1) is replaced with the requirement that the basin of attraction should have positive measure. He also requires that no smaller closed subset has the same basin, up to measure zero. This definition has withstood the test of time, and a definition of "chaotic attractor" based on it appears in [2]. Milnor's paper also provides an insightful survey of the subject and many important examples.The editors present an excellent selection of key works in the development of the statistical or measure-theoretic approach to chaotic attractors. The important work of Yorke was early in this development. To get a true feeling for his many contributions to nonlinear dynamics, a book would have to touch on almost all aspects of (low-dimensional) chaos, where he has seen new, sometimes startling, connections and in doing so has opened avenues of investigation and introduced innovative techniques.