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Re: Iterating the function f(x,y) = (sin(y)+x,sin(sin(y)+x)+y)
Posted:
Jan 4, 2014 9:20 AM


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 areapreserving, so the dynamics is pretty well described by KAMtheory. 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 KAMtheory it would be interesting to have this map embedded in a one parameter family of area preserving maps f_t where f_0 has saddelsaddle connections.
 Thomas Nordhaus



