# Mathematical Chaos - Demonstration

## The Sinai map

The Ansov or cat map takes the form:

*
*
x_{n+1} |
= |
x_{n} + y_{n} mod 1 |

y_{n+1} |
= |
x_{n} + 2 y_{n} mod 1 |

This is an area preserving map with Lyapunov exponents +/- 0.9624 and with an orbit that fills the area with uniform density (and so all dimensions *D*_{q} are 2). It is a map on the unit torus, that is uniformly stretching in the direction (x,y)=(0.85,0.53), and uniformly contracting in the orthogonal direction. The map is therefore hyperbolic, differentiable and mixing, i.e. Axiom A. This implies that it is structurally stable, and so for small enough perturbations the attractor should remain hyperbolic with capacity dimension 2.

Sinai considered the perturbation

*
*
x_{n+1} |
= |
x_{n} + y_{n} + a cos(2y_{n}) mod 1 |

y_{n+1} |
= |
x_{n} + 2 y_{n} mod 1 |

The properties for *a=0.1* are shown in the applet:

Although you can check that the box counting dimension remains 2, there is interesting structure in the probability measure, and for example the information dimension seems to be slightly below 2. (It is not known whether *a=0.1* is "small enough" e.g. to maintain the hyberbolicity.

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[Demos Introduction]
[Introduction]

Last modified Wednesday, December 15, 1999

Michael Cross