Organization of the periodic orbits in the Rössler flow

Abstract

In this paper, we systematically investigate the periodic solutions of the Rössler equations up to certain topological length. To overcome the difficulties for a return map that is multivalued and non-invertible in the nonlinear system, we propose a new approach that establishes one-dimensional symbolic dynamics based on the topological structure of the orbit. A newly designed variational method is numerically stable for cycle searching, and two-orbit fragments can be used as basic building blocks to initialize the system. The topological classification based on the whole orbit structure seems more effective than partitioning the Poincaré surface of section. The current research supplies an interesting framework for a systematic classification of periodic orbits in a chaotic flow.