On Monday, July 9, 2012 11:59:47 PM UTC+1, Richard Clark wrote: > > > > > If this were my problem, I might start with a local linearization of the operator, to determine which of the fixed points were attractive. > > > > Yes, I've looked into that; In fact neither of the fixed points are attractive. I have got somewhere with the problem. It seems that there is an attractor at infinity whose basin of attraction is the whole of the plane apart from the two fixed points and the points that go around the fixed point (1,1). Its boundary is a closed curve and points on the boundary keep going round on it. Points in the basin close to the boundary then get pulled around it an unpredictable number of times before being allowed to shoot off.
I've got further with this.
It turns out that the fixed point at (1,1) is a centre and the fixed point at (2,2) is a saddle point.
If we replace the function g by the function h, where h = 1-x if y<=1 and h = 3y-2-x if y>1 we get a piecewise linear map which has a centre at (0.5,0.5) and a saddle point at (2,2) that displays similar behaviour.
We can analyse what is happening as follows:
1. Take the triangle with vertices at (2,2) and at the points where the eigenvalues of the map that defines the top part the map cuts the line y=1.
2. Analyse what happens to the triangle when we iterate the map.
There doesn't seem to be much stuff about this out there. Does anybody know if there is a classification theorem for discrete dynamical systerms in 2 dimensions in terms of their fixed points?