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Book review

Author(s) Abraham, Ralph H.
Shaw, Christopher D.
Title Dynamics, The Geometry of Behavior, 2nd edition
Publisher Addison-Wesley
Year of publication 1992
Reviewed by Cristina Blaga

This amazing book is a successful attempt to explain one of the most modern part of science, the theory of dynamical systems, not only to a mathematician or physicist, but also to a layman with very few knowledges of mathematics (actually vector calculus and complex numbers). To reach this goal, the authors adopted the visual style. In fact, the book is an album, containing pictures and short explanations. Only at the end, in an appendix, one can find a couple of mathematical symbols. I have to say, nevertheless, that this theory is originated in physical reality, so it is quite close to intuition. I have serious doubts that such an attempt could be made for, say, the theory of categories, which is much more abstract and which appeared from necessity of mathematics itself.

The book consists of four parts (in the first edition separated in four volumes). The first part (periodic behavior) introduces the basic notions of dynamical systems theory: state spaces, dynamical system, trajectories, limit points, limit cycles, invariant tori a.o. The second part (chaotic behavior) is devoted to the different aspects of attractors (static, periodic, chaotic) as geometric models for the local asymptotic behavior of a dynamical system. The third part (global behavior) is dealing with structural stability and the generic properties related to the tangled insets and outsets. Finaly, the last part (bifurcation behaving) is an atlas of bifurcation schemes, including for instance: subtle bifurcations, fold catastrophe and other catastrophes, explosive bifurcations and fractal bifurcations. Each part of the book starts with a hall of fame, including images and short biographies of the scientists that have made important discoveries in the field. Beside these halls of fame, the book contains a great amount of historical informations about the topics touched. The pictures are very well realized and very well chosen. As I already said, I consider that this book is an incontestable success. Of course, the main responsables for this success are the authors, the first being a well known mathematician (author of Foundations of Mechanics, Benjamin, 1967), and the second a freelance filmmaker and artist.

As any respectable book, this one has too an extensive reference list (including, of course, also technical texts) and a detailed index.