Is the classical Rössler attractor periodic? A validated numerical study.

Abstract

The Rössler system is a classical low-dimensional dynamical system generating different types of attractors. The question whether the Rössler attractor observed for classical parameter values is periodic or chaotic remains an open problem. In this work, we search for periodic windows in a small neighborhood of the classical parameters. Symbolic representation of trajectories is defined and the ordering of symbol sequences is constructed with a property that for short periodic symbol sequences, their order agrees with the layout of corresponding periodic windows. Using symbolic descriptions of periodic window, the continuation technique, the bisection method, and the local exhaustive search in the symbol sequence space, we find a periodic window with a distance smaller than 2 × 10-22 from the classical case. Convergence properties of periodic attractors are studied numerically.