> > 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.