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Good introduction to the beginning student
on August 10, 2001
This book gives a quick and elementary introduction to the field of chaotic dynamical systems that could be read by anyone with a background in calculus and linear algebra. The approach taken by the author is very intuitive, lots of diagrams are used to illustrate the major points, and there are many useful exercises throughout the book. It could serve well in an undergraduate mathematics course in dynamical systems, and in a physics undergraduate course in advanced mechanics. The author emphasizes the mathematical aspects of dynamical systems, and readers will be well prepared after finishing it to read more advanced books on dynamical systems.
Chapter 1 introduces one-dimensional dynamics, with the analysis of the quadratic map given particular attention. Called the logistic map in some circles, this very important dynamical system has been the subject of much study, and exhibits generically the properties of chaotic dynamical systems. The author also gives a brief review of some elementary notions in calculus needed for the chapter, making the book even more accessible to a wider readership. The important concept of hyperbolicity is discussed in the context of one-dimensional maps and a good discussion is given on symbolic dynamics. Structural stability, which is really useful only in dynamical systems in higher dimensions, is treated here. The intuition gained in one-dimension is invaluable though before moving on to higher-dimensional examples. Sarkovskii's theorem, which states that a one-dimensional dynamical system with a period three periodic orbit has periodic orbits for all other periods, is proved in detail. In addition, the Schwarzian derivative, so important in complex dynamics, is defined here. The author also gives an introduction to bifurcation theory, which again, is most interesting in high dimensions, and introduces the concept of homoclinicity in this discussion. Maps of the circle and the all-important Morse-Smale diffeomorphisms, are treated in this chapter also. The author introduces the reader briefly to the idea of genericity when discussing Morse-Smale diffeomorphisms. Kneading theory, so important in the mathematical theory of dynamical systems, is introduced here also.
In chapter 2, the author generalizes the results to higher dimensions, and begins with a review of linear algebra and some results from multivariable calculus, such as the implicit function theorem and the contraction mapping theorem. This is followed by a treatment of the dynamics of linear maps in two and three dimensions. Whereas the canonical example of one-dimensional dynamics is represented by the logistic map, in higher-dimensional dynamics this is represented by the Smale horseshoe map. The author carefully constructs this map and details its properties. Then he takes up the hyperbolic toral automorphisms (or Anosov systems as they are called in some books). Both the Smale horseshoe map and the toral automorphisms are excellent, easily understandable examples of higher dimensional dynamics and the associated symbolic dynamics.
The concept of an attractor is also treated in chapter 2 in the context of the solenoid and the Plykin attractor. Both of these are of purely mathematical interest, but by studying them the physicist reader can get a better understanding of what to look for in actual physical examples of attractors (or the more exotic concept of a strange attractor). The author also gives a proof of the stable manifold theorem in dimension two. This is the best part of the book, for this theorem is rarely proved in textbooks on chaotic dynamics, the proof being delegated to the original papers. However, the proof in these papers is extremely difficult to get through, and so the author has given the reader a nice introduction to this important result, even though it is done only in two dimensions. This is followed by a very understandable discussion of Morse-Smale diffeomorphisms. In addition, the author introduces the Hopf bifurcation, of upmost importance in applications, and introduces the Henon map as an application of the results obtained so far.
The last chapter of the book is a brief overview of complex analytic dynamics. Complex dynamical systems are very important from a mathematical point of view, and they have fascinating connections with number theory, cryptography, algebraic geometry, and coding theory. The author reviews some elementary complex analysis and then reintroduces the quadratic maps but this time over the complex plane instead of the real line. The Julia set is introduced, and the reader who has not seen the computer graphical images of this set should peruse the Web for these images, due to their beauty. The geometry of the Julia set and the associated complex polynomial maps are given a fairly detailed treatment by the author in the space provided.