A lattice refinement scheme for finding periodic orbits

Abstract

A lattice refinement scheme based on the principle of linearized stability is introduced to locate periodic orbits in a two-dimensional map. The method locates all periodic orbits of a specified order within a given starting window and it can be equally well applied when the map is only known implicitly, e.g., as a two-dimensional surface of section arising from a three-dimensional flow. Periodic orbits in the Henon Map, the Predator-Prey Map, the Rossler Flow, and the Lorenz Flow are constructed as illustrations of the method.