Arnold's Cat Map

Arnold's cat map is a simple discrete system that stretches and folds points (, ) to (, ) mod 1 in phase space. Typically, any two points that initially are very close together quickly become separated from each other after repeated applications of the map. The picture first shears apart and later looks random, or, in some steps, shows a ghost-like, repeated image of the original. Finally, the points return to their original position after 150 steps. The picture is a portrait of Henri Poincaré, an early pioneer in the study of this kind of recurrent behavior.