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An Introduction To Chaotic Dynamical Systems

Robert L. Devaney
CRC Press
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Bill Satzer
, on
This text on chaotic dynamics is the third edition of a book whose widely used second edition is thirty years old. This book, in contrast to another of Devaney’s well-known books (reviewed here), is aimed at graduate students and upper-level undergraduate students in mathematics. It is fully a mathematics book with careful definitions, theorems and proofs.
The book is divided into three parts as in the earlier edition, but the order of the parts has been changed. The first introduces one-dimensional dynamics, but the second is now an expanded treatment of complex dynamics. The final part addresses higher dimensional dynamics. 
The first part of the book is the longest. It introduces essentially all the main ideas of nonlinear dynamics in one-dimension, with the real line and the circle. In so doing the author minimizes algebraic and topological issues and is able to focus on new concepts and techniques. Throughout the book only discrete dynamical systems are considered. With a minimum of fuss the student first learns about chaos, followed by the big topics that include structural stability, bifurcation theory, symbolic dynamics and the shift map. By the end of the first part, a student will have seen essentially all the critical concepts of dynamical systems theory.
The treatment of complex dynamics concentrates on quadratic functions in the complex plane. This new edition adds a discussion of rational mappings including singular perturbations of a polynomial, as well as an expanded treatment of the Mandelbrot set. As in the earlier editions, discussions of the Julia set and exponential mappings are also included.
The third part, which is independent of the second, discusses higher-dimensional dynamics. It explores topics like the horseshoe map of Smale, hyperbolic toral automorphisms, the stable and unstable manifold theorems, the Hopf bifurcation, and the Henon map.
This is a strong book, carefully written with a lot of thought given to the development and presentation of the subject. The author makes very good use of graphics in various forms from basic to very complex. Plenty of excellent exercises are presented. The prerequisites become more important as the book proceeds. These include a solid background in calculus, basic linear algebra, some topology, a bit of complex analysis and results from analysis like the implicit function theorem. Appendices that fill in some of these are provided.
Another book – with roughly the same subject – is Strogatz’s Nonlinear Dynamics and Chaos. It takes a much different approach, is intended for newcomers to dynamical systems, and requires only calculus with basic differential equations. Strogatz proves very little and emphasizes applications. Devaney’s book includes essentially no applications, and his focus on discrete equations is in contrast to Strogatz’s broader look at both continuous and discrete systems. The books have quite a different feel. Strogatz is more broadly addressed to mathematics, science and engineering readers, while Devaney is focused on mathematics students.


Bill Satzer ([email protected]), now retired from 3M Company, spent most of his career as a mathematician working in industry on a variety of applications. He did his PhD work in dynamical systems and celestial mechanics.