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Invitation to Dynamical Systems

Edward R. Scheinerman
Publisher: 
Dover Publications
Publication Date: 
2012
Number of Pages: 
373
Format: 
Paperback
Price: 
26.95
ISBN: 
9780486485942
Category: 
Textbook
[Reviewed by
Luiz Henrique de Figueiredo
, on
05/12/2012
]

This is Dover edition of the 1996 original, with the same nice typesetting. The book is also available in standard LaTeX typesetting at the author’s web site, which also contains a solution guide.

The book is true to its title: the aim is to take dynamical systems to a wide audience of students in engineering, science, economics, computer science, mathematics, etc. It has very few prerequisites: calculus and linear algebra, but no differential equations. The focus is on ideas, not on formal theorems and proofs.

In all, I think the author succeeds quite well in his goal. Certainly, any student that reads this book seriously will learn a lot of non-trivial interesting mathematics. Indeed, while the prose is quite agreeable, this is not a watered-down presentation. It discusses advanced results, such as Liapunov functions, the Poincaré-Bendixson theorem, symbolic dynamics, Sharkovskii’s theorem, and even proves a special case of Li and Yorke’s theorem that “Period Three Implies Chaos,” a famous Monthly article. (There are not many references, though. While this might have been a hindrance in 1996, today it’s just fine: searching the web for these results is quite easy and will probably lead the curious student to further results.)

I have only two minor quibbles: Chapter 5 on fractals, while nicely done, breaks the flow of presentation, from real quadratic dynamical systems in Chapter 4 to complex ones in Chapter 6. This last chapter is also too short, when compared with the others. Of course, complex dynamical systems is a huge subject and there are plenty of excellent introductory books, such as Devaney’s Chaos, Fractals and Dynamics: Computer Experiments in Mathematics. But these are just two minor points. The book is very nice. I’ve enjoyed reading it and I’m positive that others will as well.


Luiz Henrique de Figueiredo is a researcher at IMPA in Rio de Janeiro, Brazil. His main interests are numerical methods in computer graphics, but he remains an algebraist at heart. He is also one of the designers of the Lua language.

Preface
1. Introduction
2. Linear Systems
3. Nonlinear Systems 1: Fixed Points
4. Nonlinear Systems 2: Periodicity and Chaos
5. Fractals
6. Complex Dynamical Systems
Appendix A. Background Material
Appendix B. Computing
Bibliography
Index

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