Periodic solutions and invariant torus in the Rössler System

Abstract

The Rössler System is characterized by a three-parameter family of quadratic 3D vector fields. There exist two one-parameter families of Rössler Systems exhibiting a zero-Hopf equilibrium. For Rössler 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 orbit that bifurcates from the zero-Hopf equilibrium. To the best of our knowledge, up to now, a torus bifurcation had only been numerically indicated for the Rösler System. For Rössler 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 the known results so far regarding periodic solutions for such a family. In addition, the stability properties of the periodic solutions and invariant torus are analyzed.