Templates and subtemplates of R\"ossler attractors from a bifurcation diagram

TL;DR

This paper analyzes the bifurcation diagram of the R"ossler system, revealing that all attractors can be described by a single topological template, providing a unified symbolic dynamic framework.

Contribution

It demonstrates that a unique topological template captures the structure of all attractors in the R"ossler bifurcation diagram, unifying their topological description.

Findings

All attractors are subtemplates of a single template.

A topological partition of the bifurcation diagram is established.

The symbolic dynamics of attractors are described by one template.

Abstract

We study the bifurcation diagram of the R\"ossler system. It displays the various dynamical regimes of the system (stable or chaotic) when a parameter is varied. We choose a diagram that exhibits coexisting attractors and banded chaos. We use the topological characterization method to study these attractors. Then, we details how the templates of these attractors are subtemplates of a unique template. Our main result is that only one template describe the topological structure of height attractors. This leads to a topological partition of the bifurcation diagram that gives the symbolic dynamic of all bifurcation diagram attractors with a unique template.