Attractors, strange and perverse. (English) Zbl 0679.58029

Williams, R. F.

Differential geometry and differential equations, Proc. Symp., Shanghai/China 1981, 473-495 (1984).

[For the entire collection see Zbl 0646.00010.]
The author addresses some aspects of the “history” of strange attractors. His approach, personal in outlook, is geometric and mostly devoted to the “Lorenz attractor”.
The author considers the equations of the Lorenz model and reminds of their properties as found by Lorenz working on a computing machine: regardless of the initial conditions, the solutions proceed into the central part of space and then begin revolving around two points; the number of times a solution winds around one side before going to the other is random. The paper is devoted to expose the efforts made in order to develop an analytic model for the Lorenz attractor, suited to reflect these properties: in fact, this is the field in which the author has very significant contributions (see the references at the end of this review). The author remarks that although it is comparatively easy to formulate “computer experiments” concerning strange attractors, it is hard to prove anything analytically about them.
References: J. Birman and the author, Topology 22, 47-82 (1983; Zbl 0507.58038); Contemp. Math. 20, 1-60 (1983; Zbl 0526.58043); M. Shub and the author, “Strong stable foliation for the Lorenz attractor” (to appear); the author, Global Analysis, Proc. Symp. Pure Math. 14, 341-361 (1970; Zbl 0213.504); ibid. 14, 329-334 (1970; Zbl 0213.503); Publ. Math., Inst. Haut. Etud. Sci. 43, 169-203 (1974; Zbl 0279.58013); ibid. 50, 73-99 (1979; Zbl 0484.58021); “Lorenz knots are prime” (submitted for publication).

Reviewer: D.Savin

MSC:

37D45

Strange attractors, chaotic dynamics of systems with hyperbolic behavior

Keywords:

strange attractors; Lorenz attractor; analytic model

Citations:

Zbl 0646.00010; Zbl 0507.58038; Zbl 0526.58043; Zbl 0213.504; Zbl 0213.503; Zbl 0279.58013; Zbl 0484.58021