Topological origins of a bi-parameter periodicity hub for the R\"ossler attractor

Abstract

We explore the dynamical and topological characteristics of the R\"ossler system that lead to the existence of a periodicity hub and nested spiral in codimension-2 parameter space. We find that the hub shape is a consequence of conjugacy between the R\"ossler system and unimodal maps in the two-branch region of the parameter space. The nested spiral structure is a consequence of a topological feature of the R\"ossler system that has not been noted previously. We outline this mechanism and detail the spiral transition for the symbolic sequence of orbits up to period seven.