Dynamical systems and ergodic theory Definition

Strange attractor

Definition

Let $f \colon X \to X$ be a continuous map of a metric space, or a flow on a metric space, and let $A$ be an attractor of it. The attractor $A$ is strange if it has a fractal structure. In the most common precise form, this means that the Hausdorff dimension of $A$ , or its box-counting dimension, is not an integer.

Remarks

There is no single agreed definition. Many authors use strange attractor and chaotic attractor as synonyms, and then they ask for sensitive dependence on initial conditions on $A$ instead of, or in addition to, fractal structure. The two conditions are different: there are chaotic attractors that are not fractal, and there are fractal attractors on which nearby orbits do not separate.

Example: the Smale solenoid map doubles the angle $\theta \mapsto 2\theta$ around a solid torus and contracts strongly in the two transverse directions. Its attractor is locally the product of an interval and a Cantor set. So its dimension lies strictly between $1$ and $2$ .

Non-example: for the flow of $r' = r(1 - r^2)$ , $\theta' = 1$ in polar coordinates in the plane, the unit circle is an attractor. It is a smooth closed curve of Hausdorff dimension $1$ , an integer, so it is not strange.

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Attractor concept
Hausdorff dimension concept
Strange attractor definition

Tag	`TY6TPW88MX`
URI	`m2://TY6TPW88MX`
Kind	definition
Writing	W0 stub
MSC	37D45, 37C70, 34C28
Defines	Strange attractor
Uses	Attractor, Hausdorff dimension

Strange attractor

Definition

Remarks

Reading path

Further reading

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