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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 4, 2014 9:20 AM
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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.)

That's nice.
Fixed points are centres if m+n is odd. Eigenvalues 1/2 +- i*sqrt(3)/2.
Saddles if m+n is odd. Eigenvalues -3/2 +- sqrt(5)/2. Correct?
f is area-preserving, so the dynamics is pretty well described by
KAM-theory. I don't see many surprises - but still a fun map to iterate
(I'm going to do that).

P.S. In the context of KAM-theory it would be interesting to have this
map embedded in a one parameter family of area preserving maps f_t where
f_0 has saddel-saddle connections.

Thomas Nordhaus

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