Chaotic evolution and strange attractors. The statistical analysis of time series for deterministic nonlinear systems. Notes prepared by Stefano Isola from the ‘Lezioni Lincee’ (Rome, May 1987).

*(English)*Zbl 0683.58001
Lezioni Lincee. Cambridge etc.: Cambridge University Press. xi, 96 p. £25.00/hbk; $ 39.50/hbk; £8.95/pbk; $ 12.95/pbk (1989).

This book is a review on those aspects of dynamical systems which are more closely related with ergodic theory, namely with the study of the properties of the invariant measures generated by the time evolution themselves. The text is divided into two parts. The concept of the chaos for deterministic systems through a survey of some “historical topics” is developed in part one, like, for example, the interpolation of hydrodynamical turbulence. Some definitions of geometrical notions related to chaotic phenomena, like those of strange attractors, fractal dimensions, reconstruction of the dynamics from a time series and so on, are also given. In the second part the concept of invariant probability measure is introduced, along with some ergodic quantities such as characteristic exponents, entropy, dimensions, resonancy, etc. Also, a number of examples and figures in such a way as to give some references useful both as clarifying elements and to emphasize the deep mathematical ideas which permeate the theory of differential dynamical systems are considered. The book is an excellent introduction in chaotic evolution systems and strange attractors; for a broad audience of graduate students and faculty members.

Reviewer: L.G.Vulkov

##### MSC:

58-02 | Research exposition (monographs, survey articles) pertaining to global analysis |

37A99 | Ergodic theory |

37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |

##### Keywords:

dynamical systems; ergodic theory; invariant measures generated by the time evolution; chaos; hydrodynamical turbulence; strange attractors; fractal dimensions; probability measure; ergodic quantities; characteristic exponents; entropy; resonancy
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\textit{D. Ruelle}, Chaotic evolution and strange attractors. The statistical analysis of time series for deterministic nonlinear systems. Notes prepared by Stefano Isola from the `Lezioni Lincee' (Rome, May 1987). Cambridge etc.: Cambridge University Press (1989; Zbl 0683.58001)