Cat Map

Arnold's cat map is a simple discrete system that stretches and ``folds'' the trajectories in phase space, which is another typical feature of chaotic processes. The phase space for this simple system can be represented by a square, and the stretching and folding process is more apparent if we placed a picture of a cat in the square. One can then see the time evolution of the system by observing how the cat gets stretched, cut up, and then placed back into the square. From the figure below, we observe that typically, any two points that are initially very close together quickly become separated from each other after repeated applications of the map.

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.

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.

Here's a couple animated examples of the catmap:

This one is mapped on the Torus as opposed to a planar image:

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