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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




Iterating the function f(x,y) = (sin(y)+x,sin(sin(y)+x)+y)
Posted:
Jan 4, 2014 7:44 AM


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.)



