By C. Taylor

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20) is any member of a class of chaotic maps of the square into itself. I. V. Anosov, Sinai, and Novikov. He lectured there, and spent a lot of time with Anosov. He suggested a series of conjectures, most of which Anosov proved within a year. It was Anosov who showed that there are dynamical systems for which all points (as opposed to a nonwandering set) admit the hyperbolic structure, and it was in honor of this result that Smale named them Axiom–A systems. In Kiev Smale found a receptive audience that had been thinking about these problems.

Now, let us start ‘gently’ with chaotic dynamics. Recall that a dynamical system may be deﬁned as a deterministic rule for the time evolution of state observables. 5) and iterative maps in which time is discrete: x(t + 1) = g(x(t)), (x, g ∈ Rn ). 6) In the case of maps, the evolution law is straightforward: from x(0) one computes x(1), and then x(2) and so on. 1 Basics of Attractor and Chaotic Dynamics 21 t > 0 [Ott93]. 14. Fig. 14. Examples of regular attractors: ﬁxed–point (left) and limit cycle (right).

Sinai, who was the ﬁrst to show that a physical billiard can be ergodic, knew Krylov’s work well. On the other hand, the work of Lord Rayleigh also received vigorous development. It prompted many experiments and some theoretical development by B. Van der Pol , G. Duﬃng, and D. Hayashi . They found other systems in which the nonlinear oscillator played a role and classiﬁed the possible motions of these systems. This concreteness of experiments, and the possibility of analysis was too much of temptation for M.