Arnold's Cat


Arnold's cat, invented by Vladimir Arnold, shows the time evolution of an image in the phase space of a dynamical system. The discrete map corresponds to fixed time intervals after which the image is observed. (The basic geometry being that of a torus, the matrix has to satisfy the condition of volume preservation: det(A)=1) Since the points of the image are finite and the map is discrete, the image will eventually return to its original form. This illustrates the recurrence theorem of Henri Poincaré, which states that any dynamical system will return arbitrary close to its initial state after a certain time, thus apparently contradicting laws involving entropy and even more contradicting our everyday experience that any moment, once passed, is irretrievable.
Some examples
( 1 1 )
1 0
recurrence time 600
( 75 1 )
1 0
recurrence time 6
( 150 1 )
1 0
recurrence time 4
( 300 1 )
1 0
recurrence time 2
( 4 5 )
3 4
recurrence time 300
( 1 499 )
1 500
recurrence time 12
( 10 499 )
1 50
recurrence time 20


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Contact: higobreitenbach@arcor.de

Last modified: Sep. 2013