Dynamical Systems with Applications using MATLAB®

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Dynamical Systems with Applications using MATLAB®

Stephen Lynch
Birkhäuser (2004), 462 pp., Price: £39.99 (softcover)
ISBN 978-0-8176-4321-8.
Reviewer: Alois Steindl
Institute for Mechanics and Mechatronics
Vienna University of Technology
Vienna, Austria

This book is a follow-up to "Dynamical Systems with Applications using Maple" by the same author. It covers the same topics, which range from simple discrete dynamical systems, like the tent map, to more advanced systems like linear and nonlinear differential equations, Poincaré maps, local and global bifurcations, neural networks and Hilbert's 16th problem.

The different chapters start with an introduction to the topic and give a list of references. Furthermore, they contain a section with short MATLAB routines, which show possible treatments of the simple typical examples by MATLAB methods.

Unfortunately, the treatment of the major part of the material is quite cursory. For most dynamical systems only bifurcation diagrams from simple simulations or phase portraits are displayed.

Especially in the chapter about local bifurcation theory I would expect some hints about center manifolds or Normal Form Theory. Also the Hopf bifurcation is stated in its simplest form. The author should definitively show some real world problem which bifurcates to periodic solutions. The pitchfork diagrams in Chapter 12 (e.g. Figures 12.2, 12.6, 12.8) are also a nuisance because they show a sharp corner at the bifurcation point where the bifurcating branch has a vertical tangent. Since at the end of the chapter there is a small MATLAB function which draws these diagrams, the author should have noticed the difference. There should also be some hint to the Matcont program which treats bifurcation problems very comfortably, or to any other software package for this kind of problems.

My second criticism concerns the MATLAB part. Since MATLAB is directed more towards numerical calculations, while Maple's main strength are symbolic calculations, I would have hoped that the numerical aspects and possibilities are treated more intensively. I would also really have liked a closer connection between the main text and the program examples. There could be some section in each chapter, where the author elaborates how MATLAB could be used to treat the problems at hand. Also, the worked out programs at the end are quite simple and do not give a proper impression of MATLAB's capabilities. There are also almost no comments, which is reasonable for these problems, but the educational value would be significantly larger if the treatment of harder problems would be demonstrated by well-documented code snippets.

The main merits of the book are the well understandable introduction to different topics in dynamical systems and the presentation of quite a large number of applications, together with a well selected bibliography.

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