Arnold's cat map

From Wikipedia, the free encyclopedia

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.^[1]

Thinking of the torus as Arnold's cat map is the transformation given by the formula

Equivalently, in matrix notation, this is

That is, with a unit size equal to the width of the square image, the image is sheared one unit to the right, then one unit up, and all that lies without that unit square is wrapped around on the other respective side to be within it.

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

[edit] Properties

• Γ is invertible because the matrix has determinant 1 and therefore its inverse has integer entries,

• Γ is area preserving,

• Γ has a unique hyperbolic fixed point (the vertices of the square). The linear transformation which defines the map is hyperbolic: its eigenvalues are real numbers, one greater and the other smaller than 1, so they are associated respectively to an expanding and a contracting eigenspace which are also the stable and unstable manifolds. The eigenspace are orthogonal because the matrix is symmetric. Since the eigenvectors have rationally independent components both the eigenspaces densely cover the torus. Arnold's cat map is a particularly well-known example of a hyperbolic toral automorphism, which is an automorphism of a torus given by a square unimodular matrix having no eigenvalues of absolute value 1.^[2]

• The set of the points with a periodic orbit is dense on the torus. Actually a point has a periodic orbit if and only if its coordinates are rational.

• Γ is topologically transitive (i.e. there is a point whose orbit is dense)

• The points with period n are exactly λ₁ⁿ + λ₂ ^− n−2 (where λ₁ and λ₂ are the eigenvalues of the matrix)

• Γ is ergodic and mixing,

• Γ is an Anosov diffeomorphism and in particular it is structurally stable.

[edit] The discrete cat map

From order to chaos and back. Sample mapping on a picture of 150x150 pixels. The numbers shows the iteration step. After 300 iterations arriving at the original image

Sample mapping on a picture of a pepper. Be patient!

It is possible to define a discrete analogous 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 picture to the right, 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—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 q_t (0 ≤ q_t < N) to site q_t+1 on a circular ring with circumference N, according to the second order equation:

q_t+1 - 3q_t + q_t-1 = 0 mod N

Defining the momentum variable p_t = q_t - q_t-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:

q_t+1 = 2q_t + p_t mod N

p_t+1 = q_t + p_t 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:

q_t-1 = 2q_t - p_t mod N

p_t-1 = -q_t + p_t 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.^[3]

For an image, the relationship between iterations could be expressed as follows:

n = 0: T⁰(x,y) = Input Image (x,y)

n = 1: T¹(x,y) = T⁰(mod(2x+y,N),mod(x+y,N))

.

.

.

n = k: T^k(x,y) = T^k-1(mod(2x+y,N),mod(x+y,N))

.

.

.

Output Image(x,y) = Tⁿ(x,y)

[edit] See also

Mathematics portal

• List of chaotic maps
• Recurrence plot

[edit] References

1. ^ *(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. 
2. ^ Franks, John M. Invariant sets of hyperbolic toral automorphisms. American Journal of Mathematics, Vol. 99, No. 5 (Oct., 1977), pp. 1089–1095
3. ^ Period of a discrete cat mapping , Freeman J. Dyson and Harold Falk, American Mathematical Monthly 99 603-614 (1992).

[edit] External links

• Weistein, Eric W., "Arnold's Cat Map" from MathWorld.
• A description and demonstration, using an image of the Earth as an example
• Effect of randomisation of initial conditions on recurrence time
• Arnold's Cat Map by Enrique Zeleny, The Wolfram Demonstrations Project.
• Arnold's Cat Map, David Arnold.