A study on strange dynamics of a two-dimensional map

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