Manifold structures of unstable periodic orbits and the appearance of periodic windows in chaotic systems.

Manifold structures of the Lorenz system, the Hénon map, and the Kuramoto-Sivashinsky system are investigated in terms of unstable periodic orbits embedded in the attractors. Especially, changes of manifold structures are focused on when some parameters are varied. The angle between a stable manifold and an unstable manifold (manifold angle) at every sample point along an unstable periodic orbit is measured using the covariant Lyapunov vectors. It is found that the angle characterizes the parameter at which the periodic window corresponding to the unstable periodic orbit finishes, that is, a saddle-node bifurcation point. In particular, when the minimum value of the manifold angle along an unstable periodic orbit at a parameter is small (large), the corresponding periodic window exists near (away from) the parameter. It is concluded that the window sequence in a parameter space can be predicted from the manifold angles of unstable periodic orbits at some parameter. The fact is important because the local information in a parameter space characterizes the global information in it. This approach helps us find periodic windows including very small ones.