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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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Thomas Nordhaus

Posts: 433
Registered: 12/13/04
Re: Iterating the function f(x,y) = (sin(y)+x,sin(sin(y)+x)+y)
Posted: Jan 5, 2014 12:32 PM
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Am 04.01.2014 13:44, schrieb Richard Clark:
> 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.)


Your map indeed shows the classical behaviour described in KAM-theory
(after Kolmogorov, Arnold and Moser). I plotted a few orbits on the
torus, one of them starting near the saddle and exhibiting chaotic
(random-like) behaviour:
https://dl.dropboxusercontent.com/u/15892273/torusmap.jpg

--
Thomas Nordhaus



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