The Experiment

        Having developed the mathematical tools to address the heart of this discussion, an experiment may be performed to determine if, indeed, there exists any order – underlying or otherwise – to the chaotic mapping of Arnold’s cat map.  For this purpose, a program was developed for MATLAB by the author, which is available in the self-extract ZIP file here.  Simply unzip this files into a directory on your MATLAB path and type "catbox."  The following 124 x 124 image of the Earth was iterated with the Arnold's Cat map transformation.  Included is an animation showing the iteration process, where each cycle starts anew with the image fading in.

     
     
     
        The pixels rapidly degenerate into a television-static of chaos by iteration number five, with some unintelligible order prominent in a number of iterations prior to the original image reappearing on the fifteenth iteration.  Therefore, the image is said to have a period of fifteen. That periodicity should be observed in a system such as this is quite extraordinary.  It’s not unlike a coherent image spontaneously leaping out of the salt-and-pepper of television static, a cogent message spoken from the howl of radio static – loaded dice, indeed!

        Experimentally, no elegant model could be developed for the relationship between the period of an image and n, its number of rows or columns.  In general, it may be claimed that as the value of n increases, the period tends to increases. However, this is not always true.  For example, a 101 x 101 image has a period of twenty-five; whereas, a 124 x 124 image, as we just learned, has a period of fifteen.  Other luminaries have found a relationship where this experimenter failed – but it certainly cannot be claimed to be elegant nor robust .  Let the period be  .
     

    1. (n) = 3n if and only if n = 2*5k for k  = 1,2,... 
    2. (n) = 2n if and only if n = 5k or n = 6*5k for k = 1,2,...
    3. (n)  12n/7 for all other choices of n
     
        Let us address that ageless question of Mankind: Why?  Why does order emerge out of this apparently chaotic mapping?  The best approach to answer this question is, perhaps, to examine the behavior of a single pixel, to determine what effect Arnold’s cat map has upon it.  Consider the ordinary pixel ( 52,13) of the 124 x 124 image considered previously.  It takes the following path:
     
     
    After fifteen iterations, the pixel – as would any other pixel in the image – has returned to its initial position, and it would continue eternally along this circle if iterated accordingly.  This agrees with the earlier observation that the 124 x 124 image has a period of fifteen.