Trajectories associated with parabolic and hyperbolic periodic points of piecewise linear area-preserving map

Abstract

This paper investigates the characteristics of trajectories and invariant sets associated with parabolic and hyperbolic periodic points of a piecewise linear area-preserving map. When the fixed points in both partitions are parabolic, the trajectories may diverge or rotate repeatedly around the origin. By analyzing the trajectory motion in different regions of the phase plane, the characteristics of the trajectory repeated rotation around the origin are shown, and the trajectories may be either divergent or periodic. The necessary conditions for generating stable periodic orbits are presented. It is proved that there are forward invariant sets and that the trajectories from the invariant sets diverge in different quadrants and along a family of invariant lines parallel to the diagonal bisector of the quadrant. We reveal why there are island chains with invariant line segments in a chaotic sea. The method for determining invariant line segments is presented. For the map with two hyperbolic fixed points in both partitions, it is proved that there are forward invariant sets in the phase plane and all trajectories from the invariant sets diverge along a family of hyperbolic invariant sets.