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Re: Iterating the function f(x,y) = (sin(y)+x,sin(sin(y)+x)+y)
Posted:
Jan 4, 2014 7:25 PM


Am 04.01.2014 23:38, schrieb quasi: > quasi wrote: >> Richard Clark wrote: >>> Thomas.Nordhaus wrote: >>>> Thomas.Nordhaus wrote: >>>>> Richard Clark wrote: >>>>>> >>>>>> 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 ... >>>>> >>>>> 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 >>>> >>>> Actually, your map f lifts to a map on the torus if you look >>>> at x and y modulo 2*Pi. The map then only has two centres at >>>> (0,Pi) and (Pi,0) and two saddles at (0,0) and (Pi,Pi). Still >>>> fun to play around with. >>> >>> Yes, it can be reduced mod 2*pi but it's worth using the >>> original to demonstrate the random walk. >>> >>> I don't think the map is area preserving  As evidence for this >>> I would point out that two points close to (0,0), say (0.1,0) >>> and (0.11,0), can end up very far apart if we iterate the >>> function enough times. >> >> The function f is definitely area preserving on R^2  check the >> Jacobian. > > Of course, in this context, area preserving means _locally_ > area preserving. > > The function f is definitely not globally area preserving. > > After all, f is periodic.
No, f isn't periodic  it is a diffeomorphism! You can solve the equations
(1) X = sin(y) + x, (2) Y = sin(sin(y) + x) + y uniquely for x and y: (2),(1) => y = Y  sin(X) and then (1) => x = X  sin(Ysin(X)). Therefore:
f^(1)(X,Y) = (X  sin(Ysin(X)) , Y  sin(X)).
So f ia locally and globally areapreserving.
> > quasi >
 Thomas Nordhaus



