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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

 Messages: [ Previous | Next ]
 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 9:57 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.

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.

--
Thomas Nordhaus

Date Subject Author
1/4/14 Richard Clark
1/4/14 Thomas Nordhaus
1/4/14 Thomas Nordhaus
1/4/14 Thomas Nordhaus
1/4/14 Richard Clark
1/4/14 quasi
1/4/14 quasi
1/4/14 Richard Clark
1/4/14 quasi
1/4/14 Thomas Nordhaus
1/5/14 quasi
1/5/14 Thomas Nordhaus
1/5/14 Richard Clark