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

TL;DR

This paper investigates the topological and dynamical features of the R"ossler system that give rise to a periodicity hub and nested spiral structures in parameter space, revealing new topological insights.

Contribution

It uncovers the topological origins of the periodicity hub and nested spiral in the R"ossler system, linking them to conjugacy with unimodal maps and novel topological features.

Findings

The shape of the hub results from conjugacy with unimodal maps.

Nested spiral structure is due to a previously unnoted topological feature.

Detailed the spiral transition for orbits up to period seven.

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.