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Chaos theory tamed. (English) Zbl 1087.37500
London: Taylor and Francis (ISBN 0-7484-0749-9/hbk). xviii, 499 p. (1997).
There have been written many books on chaos so far. But to write a good one on this rather difficult interdisciplinary subject is a very hard task. So a quality of books on chaos differs a lot. According to the author the reviewed book “intends to bridge the gap between non-mathematical popular treatment and the highly technical mathematical publications”. So the goal of author was “to present a basic, semitechnical introduction to chaos”. How far he was successful it will be analyzed here. The book is divided into seven parts. The Part I presents some background for chaos theory. Then in Part II, the auxiliary toolkit – namely the phase space, vectors, probability and information, autocorrelation, and Fourier analysis – is introduced. The Part III has a literal title “How to get there from here ”. Then, some characteristics of chaos are mentioned in a very noncomplete manner (Part IV). Phase space signatures, Part V is very short and mainly concerns attractors reconstruction. The problem of different kinds of dimensions is treated in Part VI. The last part VII is devoted to quantitative measures of chaos, namely Lyapunov exponents, Kolmogorov-Sinai entropy and mutual information and redundancy. The book ends with the Epilogue. Besides, a very strange Appendix – the only one – is added bringing elementary laws of power, roots, and logarithms. Unfortunatelly it contains some errors. The glossary of used terms could be of help (25 pages!) but there are many vague definitions of basic notions. References are not so much representative, not update, in fact till about 1993-4. The Index is quite detailed and convenient. This thick book (499 pages) was written by retired geologist/hydrologist by training as the author mentions in the preface. Why not? One should say. By the way a problem of turbulence (hydrology) is still one of the most difficult, unsolved problems in physics. Besides, there have been worked out a dynamic model of Earth dynamo (geology) with a chaotic behavior. Alas, there is no mention about these topics in the book, et all. The organization of the book is not very natural. E.g., only after a definition of chaotic (strange) attractor (Part IV, 15), fractals and fractal dimension are introduced later on (Part VI, 20), and so on. Besides, one can suspect the definition of strange attractor is not rigorously presented in the book (p. 227). There are some fine mathematical nuances concerning this delicate notion which the author does not respect so much. The point is that strange attractors are commonly taken as chaotic (due to a sensitive dependence upon initial conditions because of nonlinearity of the dynamics) and fractal= in this case strange (it is not a point, nor a limit cycle), with a fractal dimension because of dissipativity or restrictness in a case of Hamiltonian systems. So there are chaotic attractors which are not fractal, e.g., hyperbolic systems, and opposite is also true. The book, it seems, was written by the enthusiastic writer. But he published nothing else on chaos, at least there is no citation in the book of any authors paper. The book is readable in a belletristic sense but seems to be vague in more rigorous sense. This is a little bit troublesome anytime it happens. In a sense, the book is a kind of glossary of chaos theory in the beginning of 80s of last century.

37-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to dynamical systems and ergodic theory
37Dxx Dynamical systems with hyperbolic behavior
37Nxx Applications of dynamical systems