In mathematics, 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, hence the name. Thinking of the torus as the quotient space, Arnold's cat map is the transformation given by the formula Equivalently, in matrix notation, this is That is, with a unitequal to the width of the square image, the image is sheared one unit up, then two units to the right, and all that lies outside that unit square is shifted back by the unit until it is within the square.
It is possible to define a discrete analogue of the cat map. 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. After some iterations, the resulting image appears rather random or disordered, yet after further iterations the image appears to have further order—ghost-like images of the cat, multiple smaller copies arranged in a repeating structure and even upside-down copies of the original image—and ultimately returns to the original image. The discrete cat map describes the phase space flow corresponding to the discrete dynamics of a bead hopping from site qt 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 onto itself: 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: For real variablesq 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. For an image, the relationship between iterations could be expressed as follows:
