The particular instance of chaos we will explore in this discussion is the chaotic mapping called Arnold’s cat map in recognition of Russian mathematician Vladimir I. Arnold, who discovered it using an image of a cat. It is a simple and elegant demonstration and illustration of some of the principles of chaos – namely, underlying order to an apparently random evolution of a system. An image (not necessarily a cat) is hit with a transformation that apparently randomizes the original organization of its pixels. However, if iterated enough times, as though by magic, the original image reappears.

    If we let

be a n x n matrix of some image, Arnold’s cat map is the transformation
 

where mod is the modulo of

and n.  For example,

Since the signs of both arguments are the same sign in this exercise, the modulo will simply be the remainder of the long division of

and n.

    To better understand the mechanism of the transformation, let us decompose it into its elemental pieces.
 
 

1. Shear in the x-diection by a factor of 1.

2. Shear in the y-direction by a factor of 1.

3. Evaluate the modulo.

Included below is a visual aide illustrating these steps. The first step shows the shearing in the x-and y-directions, followed by evaluation of the modulo and reassembly of the image.