Invariant regions in piecewise linear area-preserving map

Abstract

This paper studies regular islands (invariant regions) associated with elliptic periodic points of a piecewise linear area-preserving map. We reveal why regular island chains are interspersed in a chaotic sea, which is typical of the behavior exhibited by the area-preserving map. We find that regular islands may be ellipses or polygons and this depends on whether the rotation number of the rotation matrix that defines the linear map is irrational or rational. We also present a method for determining the boundary of these regular islands and chaotic seas (the largest invariant set). We show that critical lines play an important role in determining the boundaries of regular islands and chaotic seas.