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 solution that bifurcates from the zero-Hopf equilibrium. 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 currently known results regarding periodic solutions for such a family. In addition, the stability properties of the periodic solutions and invariant torus are analysed.
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