Arnold's cat map
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In dynamical systems theory, Arnold's cat map is a chaotic map from the torus into itself, named after Vladimir Arnold, who demonstrated its effects in the 1960s using an image of a cat. One of this map's features is that image being apparently randomized by the transformation but returning to its original state after a number of steps.
Arnold's cat map is an important example of Anosov diffeomorphism. The map describes the phase space flow corresponding to the discrete dynamics of a bead hopping from site qt (0 =< qt < N) to site qt+1 on a circular ring with circumference N, according to the second order equation:
- qt+1 - 3qt + qt-1 = 0 mod N
Defining the momentum variable pt = qt - qt-1, the above second order dynamics can be re-written as a mapping of the square 0 =< q, p < N (the phase space of the discrete dynamical system) onto itself:
- qt+1 = qt + pt mod N
- pt+1 = qt + 2pt mod N
This Arnold cat mapping shows mixing behavior typical for chaotic systems. However, since the transformation has a determinant equal to unity, it is area-preserving and therefore invertible the inverse transformation being:
- qt-1 = 2qt - pt mod N
- pt-1 = -qt + pt mod N
For real variables q and p, it is common to set N = 1. In that case a mapping of the unit square with periodic boundary conditions onto itself results.
When N is set to an integer value, the position and momentum variables can be restricted to integers and the mapping becomes a mapping of a toroidial square grid of points onto itself. Such an integer cat map is commonly used to demonstrate mixing behavior with Poincaré recurrence utilising digital images. The number of iterations needed to restore the image can be shown never to exceed 3N.
[edit] See also
[edit] References
- (French) V. I. Arnold, A. Avez (1967). Problèmes Ergodiques de la Mécanique Classique. Paris: Gauthier-Villars.
- English translation: V. I. Arnold, A. Avez (1968). Ergodic Problems in Classical Mechanics. New York: Benjamin.
[edit] External links
- Arnold's cat map at the MathWorld
- A description and demonstration, using an image of the Earth as an example