A study on strange dynamics of a two-dimensional map

The nonlinear dynamics of a two-dimensional map system on a plane is studied. We found that the attractor of the system changed from stable focus, stable invariant circle(limit circle) to the chaotic attractor contracted into low-dimensional manifold with one positive Lyapunov exponent, finally to the chaotic attractor filling a zone with a smooth boundary with two positive Lyapunov exponents during the change of the system parameters. The characters of the fixed points are analyzed. We found that the unstable second class node and the unstable even period points are arranged alternatively on the boundary.