Book Review

Source: Mathematical Reviews, Jan. 1998

Chaos. (English)
An introduction to dynamical systems.

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

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

In recent years there has been a small but growing list of books intended as texts for an undergraduate course in dynamical systems in general, and chaos in particular. The present work, written by some prominent contributors to the development of the field, is a nice addition to that list. With regard to both style and content, the authors succeed in introducing junior/senior undergraduate students to the dynamics and analytical techniques associated with nonlinear systems, especially those related to chaos.

There are several aspects of the book that distinguish it from some other recent contributions in this area, such as one by R. Devaney [A first course in chaotic dynamical systems, Addison-Wesley, Reading, MA, 1992; MR 94a:58124]. First, the treatment of discrete systems here maintains a balanced emphasis between one- and two- (or higher-) dimensional problems. This is an important feature since the dynamics for the two cases and methods employed for their analyses may differ significantly. Also, while most other introductory texts concentrate almost exclusively upon discrete mappings, here at least three of the thirteen chapters are devoted to differential equations, including the Poincare-Bendixson theorem. Add to this a discussion of omega-limit sets, including periodic and strange attractors, as well as a chapter on fractals, and the result is one of the most comprehensive texts on the topic that has yet appeared.

Such an inclusion of topics does come at a price. While definitions and theorems are stated formally and in sufficient technical detail to constitute a serious introduction to the field, analytic proofs of most results are omitted. Among the few exceptions to this are proofs of the stable manifold theorem, the Poincare-Bendixson theorem, and the cascade theorem, which appear in later chapters of the book. This absence of formal proofs is at present typical of all but a few undergraduate texts in this area.

The book begins with two chapters that introduce respectively the two types of systems that are to be the primary focus of the work. Chapter 1 studies elementary concepts related to iteration of one-dimensional maps: fixed and periodic points, stability, the logistic equation, itineraries, sensitive dependence. Chapter 2 introduces analogous issues for two-dimensional maps: linear and nonlinear maps, the Jacobian matrix, eigenvalues, sinks, sources, saddles, stable and unstable manifolds.

The treatment thus far stops just short of chaos, which is the study of the next four chapters. Chapter 3 investigates chaos for one-dimensional maps. Special emphasis is given to Lyapunov exponents, the primary means used to determine chaotic orbits. Before treating chaos in two dimensions, in the next chapter the authors discuss fractals, including methods for computing the dimension. Chapter 5 resumes the discussion of chaos, in this case for two-dimensional maps. Lyapunov exponents are once again emphasized, along with some purely qualitative issues, such as fixed point theorems, shift maps, and horseshoe maps. Chaotic attractors of both one- and two-dimensional systems are examined in Chapter 6. In this context omega-limit sets and invariant measures are defined.

For the next three chapters the focus is shifted to the other type of dynamical system considered in the text, differential equations. Chapter 7 introduces one- and higher-dimensional, linear and nonlinear, continuous systems. Limit sets and periodic orbits of planar flows are the subject of the next chapter, which includes a proof of the Poincare-Bendixson theorem. Chapter 9 studies chaos for continuous systems, and considers some famous examples: the Lorenz attractor, the Rossler attractor, Chua's circuit and forced oscillators.

The next chapter returns to discrete mappings with a more careful analysis of stable and unstable manifolds, and implications for chaos of homoclinic and heteroclinic points. A proof of the stable manifold theorem also appears here.

Although previously introduced, a detailed examination of bifurcation is saved for the next two chapters. Parameterized families of one- and two-dimensional maps and of differential equations, and the types of bifurcation associated with them, are considered in Chapter 11. Cascades of period-doubling bifurcations and a proof of the cascade theorem are the subject of Chapter 12.

The final chapter is a brief discussion of how the state of a system can be reconstructed from experimental time series data. Delay plots and embedology are described.

Overall, the book is well designed and typeset, and is written in a casual, easy-to-read style with a generous assortment of clearly drawn graphs and diagrams, as well as some full color plates. Suggested exercises and computer experiments, intended to verify or further elaborate certain points, appear regularly throughout the main body of the text. These are supplemented at the end of each chapter with a full set of exercises, as well as with two special sections. The first, called a Challenge, allows the reader to develop and/or prove some related mathematical observations with the use of supplied hints. The other, called a Lab Visit, describes the results of laboratory experiments or other scientific observations that demonstrate real-world applications of concepts developed in the chapter. Answers to or hints for selected exercises are included in the back of the book, along with several appendices that review some matrix algebra and the numerical solution of differential equations.

At nearly 600 pages the text may seldom be covered in its entirety in a single semester, although the absence of detailed proofs may hasten the pace. The wide range of topics allows some flexibility. Several chapters could be partially of entirely bypassed at the discretion of the instructor.

Besides the calculus sequence, prerequisites for such a course would ordinarily include linear algebra and differential equations, making it suitable primarily for undergraduate students of mathematics, physical science, or engineering. The latter prerequisite might be eliminated if the chapters on differential equations were skipped either in the interests of time or to reach a wider audience.

In short, the book is a significant contribution to the increasing collection of texts on this topic. The authors have succeeded in taking the most important ideas from dynamical systems and chaos, and presenting them to undergraduates in a serious but accessible manner. The work should be a definite consideration for anyone contemplating an introductory course in this area.