Chaos and Stretching/Folding

Written by Paul Bourke
January 2005

In a geometric sense, stretching and folding in phase space often gives rise to chaotic behaviour. Stretching results in nearby points diverging, folding results in distant points being mixed together. The often quoted and easy to imagine example of this is a ball of dough that is repeatedly rolled out and then folded. During the rolling stage (stretching) two arbitrarily close points get separated and on each fold distant points can become neighbours.

In the following the first image is stretched as if on a rubber sheet, those parts of the image outside the initial rectangle are cut and reinserted as if the sheet was topologically a torus (the left edge joins with the right edge and the top edge joins with the bottom edge). This is a coarse simulation of a stretching and folding operation. The result often seems a scramble without any apparent order, other times the original image can be seen as phases align to give rise to a recurrence effect. Normally in a chaotic transformation this recurrence is vastly less likely to arise than here.

 0 (Original image) Iteration 1 2 37 63 90 94 95 110 186 187 Height of image = 188