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.

中文翻译：

---

Rössler系统中的周期解和不变环

Rössler系统的特征在于三参数二次方3D矢量场。存在两个展现零霍夫平衡的Rössler系统一参数系列。对于靠近其中一个系列的Rössler系统，我们提供了通用条件，以确保存在圆环分叉。在这种情况下，圆环围绕着一个周期解，该周期解从零霍夫平衡开始分叉。对于接近另一个族的Rössler系统，我们提供了从零霍夫平衡开始分叉的周期解的存在的一般条件。这改善了关于此类家庭的定期解决方案的当前已知结果。另外，分析了周期解和不变环的稳定性。