LECTURE 32: November 15

Strange Attractors

The term strange attractor appears to have been coined by Takens and Ruelle around 1971. Here is a quote from Ruelle:

I asked Floris Takens if he had created this remarkably successful expression. Here is his answer: "Did you ever ask God whether he created this damned Universe? ... I don't remember anything ... I often create without remembering it..." The creation of strange attractors thus seems to be surrounded by clouds and thunder. Anyway, the name is beautiful, and well suited to these astonishing objects, of which we understand so little.

- David Ruelle The Mathematical Intelligencer 2 126 (1980)

Stange attractors seem to be characterized by the following properties:

1. they represent the asymptotic behavior of the system, starting at any point in a continuous "basin" of possible initial conditions, after transients have died out,

2. they have fractal dimension (see pages 489-491 of the textbook),

3. the dynamics on the attractor is sensitively dependent on initial conditions, and

4. the attractor is invariant in the sense that the system comes arbitrarily close to any point on the attractor independently of the initial conditions.

Visit Professor Julian Sprott's Fractal Gallery for some cool pictures of strange attractors!

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