Arnold's cat map

In the original book it was captioned by a humorous footnote, The Société Protectrice des Animaux has given permission to reproduce this image, as well as others.In Arnold's native Russian, the map is known as "okroshka (cold soup) from a cat" (Russian: окрошка из кошки), in reference to the map's mixing properties, and which forms a play on words.

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.

As can be seen in the adjacent picture, the original image of the cat is sheared and then wrapped around in the first iteration of the transformation.

The discrete cat 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: 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: This Arnold cat mapping shows mixing behavior typical for chaotic systems.

In that case a mapping of the unit square with periodic boundary conditions onto itself results.

Such an integer cat map is commonly used to demonstrate mixing behavior with Poincaré recurrence utilising digital images.

Picture showing how the linear map stretches the unit square and how its pieces are rearranged when the modulo operation is performed. The lines with the arrows show the direction of the contracting and expanding eigenspaces

From order to chaos and back. Sample mapping on a picture of 150x150 pixels. The number shows the iteration step; after 300 iterations, the original image returns.

Sample mapping on a picture of a pair of cherries. The image is 74 pixels wide, and takes 114 iterations to be restored, although it appears upside-down at the halfway point (the 57th iteration).