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The R\"ossler System is characterized by a three-parameter family ofquadratic 3D vector fields. There exist two one-parameter families of R\"osslerSystems exhibiting a zero-Hopf equilibrium. For R\"ossler Systems near to oneof these families, we provide generic conditions ensuring the existence of atorus bifurcation. In this case, the torus surrounds a periodic solution thatbifurcates from the zero-Hopf equilibrium. For R\"ossler Systems near to theother family, we provide generic conditions for the existence of a periodicsolution bifurcating from the zero-Hopf equilibrium. This improves currentlyknown results regarding periodic solutions for such a family. In addition, thestability properties of the periodic solutions and invariant torus areanalysed.

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