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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
Re: Iterating the function f(x,y) = (sin(y)+x,sin(sin(y)+x)+y)
Posted: Jan 5, 2014 12:59 PM
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On Sunday, January 5, 2014 5:32:08 PM UTC, Thomas Nordhaus wrote:
> 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.
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> >
>
> > 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.
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> >
>
> > It therefore generates a random walk around the centres; Changing the starting point even slightly generates a completely different random walk.
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> >
>
> > (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


Thanks for posting that Thomas. It's the same as the result that I obtained, although I did it on Excel and didn't reduce it to the torus. As an orbit approaches one of the saddles it isn't possible to predict whether it will turn left or right.

I suppose the next step for me is to get some books on KAM theory!




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