UK Nonlinear News, November 2001

Dynamical Systems with Applications to Maple

By S. Lynch

Reviewed by James D. Meiss

398 pages.

This book is, as the title correctly states, a text about dynamical systems first and applications to Maple second. The text treats a remarkable spectrum of topics, but is necessarily brief on many of these. Theorems are carefully stated, but most are not proved. I used this text in a reading course last summer with a student, recommending that he use the excellent text by Perko as his primary source, and the current book for exercises and examples. The student, however, quickly found the more formal and analytical treatment of Perko too dry for independent study and concentrated on using Lynch's book. Perhaps your advanced undergraduate and beginning postgraduate students will feel the same way.

The text begins with a quick tutorial of some basic Maple commands: these are timed by the author to constitute two, one-hour sessions. As these tutorials consist only of a list of Maple commands, they are rather monotonous to go through, and I would not recommend them as a way of learning Maple from the start. However, given that you do have some familiarity with Maple, they do form a compact reference that you can use to refresh your skills.

The first portion of the book quickly presents the standard course on ODE's and the phase plane, such as is treated in much more detail in many standard texts. In the second portion of the book, chaos is introduced through a study of three-dimensional flows such as the Lorenz and Rossler models. It is here that the use of Maple becomes much more fun, as three-dimensional plots are easily constructed and can be rotated at will. Another strength of Maple is the "poincare" command, which can easily make plots of the Poincare return map. Homoclinic bifurcations, bifurcations of limit cycles, and Lienard systems are also treated here.

The final portion of the text introduces discrete dynamics. Nice Maple routines are given for drawing cobweb and bifurcation diagrams, Julia sets and the ubiquitous Mandelbrot set. A couple of chapters on fractals introduce the concepts of self-similarity, box-counting dimension and even the spectrum of dimensions for multifractals. The control of chaos is also discussed, and an algorithm for the OGY method applied the Henon map is given.

Though Maple is used throughout the text to create the figures, these techniques are not integrated with the presentation, but rather presented in the last section of each chapter. This has both pluses and minuses: the text is cleaner, but the tool is exploited less than I had expected. In the Maple sections, the necessary commands to reproduce the figures are given, but there is typically no discussion of the commands, their arguments and options. Of course, one can access the Maple help facility for this. I would have liked to see more discussion of the use of Maple --- both its strengths and weaknesses. For example there are several tantalizing remarks such as "this overcomes the problems with DEplot" that are not explained.

I would have also liked to see more exploitation of the algebraic strengths of Maple. For example the algorithms for the computation of normal forms, power series expansions for center manifolds, and averaging techniques are not included. There are a few serious computations given, for example, that of the power series for a Lyapunov function near a fine focus.

I think that this text has a little for everyone. It can serve as an introduction and refresher to many of the topics of dynamical systems, and will help even the most jaded reader, such as this reviewer, enjoy some of the interactive aspects of studying dynamics using Maple.

UK Nonlinear News thanks Birkhauser for providing a review copy of this book.

A listing of books reviewed in UK Nonlinear News is available.

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Last Updated: 31st October 2001.