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Topic: Iterating the function f(x,y) = (sin(y)+x,sin(sin(y)+x)+y)
Replies: 12   Last Post: Jan 5, 2014 12:59 PM

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Richard Clark

Posts: 17
Registered: 7/3/12
Iterating the function f(x,y) = (sin(y)+x,sin(sin(y)+x)+y)
Posted: Jan 4, 2014 7:44 AM
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Take the function f(x,y) = (sin(y)+x, sin(sin(y)+x)+y)

This has fixed points at (n.pi,m.pi) for integer values of n and m.

These are alternately centres and saddle points.

If you start an iteration of the function close to one of the centres e.g. (pi,0) you go around a loop containing the centre forever.

If, however, you start the iteration of the function close to one of the saddle
points, e.g. (0,0), an interesting thing happens: it goes around one of the adjacent centres a certain number of times, then goes starts going around a centre adjacent to the original centre a certain number of times, then starts going around a centre adjacent to THAT centre a certain number of times etc.

How many times it goes around the centre, and which of the 4 adjacent centres it then starts going around is sensitive to initial conditions.

It therefore generates a random walk around the centres; Changing the starting point even slightly generates a completely different random walk.

(I've iterated it a million times, but a few thousand is sufficient to see the effect.)

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