# Periodic solutions and invariant torus in the R\"ossler System

## TL;DR

This paper investigates the bifurcation phenomena in the R"ossler System, identifying conditions for the emergence of invariant tori and periodic solutions near zero-Hopf equilibria, and analyzing their stability.

## Contribution

It provides new generic conditions for torus and periodic bifurcations in the R"ossler System, enhancing understanding of its complex dynamics.

## Key findings

- Existence of torus bifurcation near one family of R"ossler Systems.
- Conditions for periodic solutions bifurcating from zero-Hopf equilibria.
- Analysis of stability properties of solutions and invariant tori.

## Abstract

The R\"ossler System is characterized by a three-parameter family of quadratic 3D vector fields. There exist two one-parameter families of R\"ossler Systems exhibiting a zero-Hopf equilibrium. For R\"ossler Systems near to one of these families, we provide generic conditions ensuring the existence of a torus bifurcation. In this case, the torus surrounds a periodic solution that bifurcates from the zero-Hopf equilibrium. For R\"ossler Systems near to the other family, we provide generic conditions for the existence of a periodic solution bifurcating from the zero-Hopf equilibrium. This improves currently known results regarding periodic solutions for such a family. In addition, the stability properties of the periodic solutions and invariant torus are analysed.