UK Nonlinear News, February 2001


Introduction to discrete dynamical systems and chaos

By Mario Martelli

Reviewed by Jeroen Lamb

Wiley-Interscience series in discrete mathematics and optimization, 1999.
ISBN: 0-471-31975-9 (Hardcover). 60.95

Currently there is a wealth of introductory textbooks on dynamical systems available, and the book by Martelli under review is a recent addition to this long list. The author has chosen to focus on discrete dynamical systems (iterated maps) in dimension one and two.

A good point of the text is that it presents elementary ideas at a slow pace with many examples. This makes the text in principle suitable for undergraduate students with only a basic knowledge of linear algebra and calculus. Moreover, the pace of the book and its choice of examples make it in my view very suitable for self-study.

During the course the author teaches some elementary analysis and linear algebra, with the application to dynamical systems as motivation. I like this approach and it provides the book with a backbone.

In my view, a weak point of the book is that it lacks original direction and comes across as if it was written in the early 1980s. In particular, the presented material is not always clear and coherent, reminiscent of the early textbooks in dynamics. To give an example: fractals and fractal dimensions are introduced, but it is not made clear what the connection is between fractals, chaos, and nonlinear dynamical systems. To be fair, there are many textbooks that suffer from this vagueness, and the connection is not so trivial (though worthwhile elaborating on).

The book is accompanied by a set of programs in Mathematica. This is in principle a very good initiative. Unfortunately, while reviewing the book, I did not have a copy of Mathematica on my computer, so I couldn't assess the quality of these programs. This illustrates the problem of choosing a particular computer algebra platform when writing a book.

The text provides an interesting set of course notes, with some original directions and approaches. However, I wonder whether anyone would use this textbook to teach a first course in dynamical systems. The subjective and somewhat dated approach by the author, as well as the poor layout of the text, makes this in my view unlikely.

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

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


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