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 underlying geometry is the twodimensional flat map of a torus. Since the points of the image are finite and the map is discrete, the image will eventually return to its original form. This does by no means prove but does illustrate very nicely 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 defying the laws involving entropy and even more contradicting our everyday experience that any moment once passed is irretrievable.
The matrix has to satisfy the condition of volume preservation: det(A)=1 and (in my example) the entries of the matrix have to be whole numbers. Since the screen is used as position memory, the reurrence time for each system depends on your screen resolution. Changing the browser size and restarting will lead to different results.
Further links: Kantz: Influence of Irrational Numbers;   Recurrence Plots;  

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  ⇐     slow      fast     ⇒  

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Some examples

( 1 1 )
1 2
( 1 1 )
-1 0
( 150 1 )
-1 0
( 300 1 )
-1 0
( 4 5 )
3 4
( 1 499 )
1 500
( 10 499 )
1 50

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Last modified: Mai 2018