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The essence of chaos. (English) Zbl 0835.58001

Seattle, WA: University of Washington Press. xii, 227 p. (1993).
The theory of chaos is one of the most exciting areas of research today. One of the features of a chaotic system is the sensitive dependence on initial conditions which leads to a lack of predictability of the evolution of the system. This effect was discovered by the MIT meteorologist Lorenz (the author of the book) in the late fifties, investigating numerically a system of differential equations which he had refined from mathematical models designed to make long range weather conditions.
As an invited Jessie and John Danz Lecturer, Lorenz delivered at the University of Washington in 1990 a set of lectures under the title “The Essence of Chaos”. The book contains a completed version of those lectures.
In the first part of the book, the author gives intuitive descriptions of the basic notions in dynamical system theory and chaos. In order to illustrate the fundamental properties of a chaotic behaviour, Lorenz uses the model of a board sliding down a ski slope. By means of this model he explains the concept of strange attractor and the process of bifurcation. A lot of other simple examples from every day life are given to make understandable the essential nature of chaotic systems.
In the second part of the book the author presents the atmosphere as an example of a dynamical system exhibiting an intricate behaviour and describes manifestations which entails us to consider them as chaotic manifestations. Some procedures through which the presence of chaos might be confirmed are also given, and finally, some of the consequences of the atmosphere’s chaotic behaviour are examined.
The third part contains the third lecture delivered at the University of Washington – “Encounters with chaos”, and three appendices. In this lecture a historical account on the development of the dynamical system theory is presented. The contributions of the mathematicians Poincaré, Birkhoff, Smale are especially pointed out. The story of his own experiments on the dynamics of the system known as Lorenz system is also presented.
In order to understand better the chaotic behaviour the author gives for the readers some methods of making their “own chaos”, that is to construct chaotic systems. At the end of this part nonlinearity, complexity and fractality are described as phenomena related to chaos.
The book ends with three appendices. The first one presents the Lorenz’s seminal paper “Does the flap of a butterfly’s wing in Brasil set off a tornado in Texas?”, the second gives the mathematical equations of the systems whose behaviour was described across the book, and the last is “A brief dynamical-systems glossary”.
The book is intended, as the lectures were, for a general audience. The chaos story is presented in a nontechnical language, but any mathematician, applied scientist or student will find it a very interesting and exciting history of the chaos theory as it is seen by one of its “parents”.

MSC:

37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
00A06 Mathematics for nonmathematicians (engineering, social sciences, etc.)
58-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to global analysis
37Cxx Smooth dynamical systems: general theory
58-03 History of global analysis
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
86A10 Meteorology and atmospheric physics
01A65 Development of contemporary mathematics
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