On the density of periodic windows for the Rössler system

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 study the problem whether periodic windows are dense in the parameter region close to the classical case. Symbolic representation of trajectories is defined and the ordering of symbol sequences is constructed with a property that for periodic symbol sequences their order agrees with the layout of corresponding periodic windows. Using symbolic descriptions of periodic window, the continuation technique and the bisection method we find sinks existing for parameter values at a distance smaller than 10⁻¹⁵ from the classical case. Convergence properties of periodic attractors are studied numerically.