In recent publications [Llibre, 2014; Llibre & Makhlouf, 2020], time-averaging method was applied to studying periodic orbits bifurcating from zero-Hopf critical points of two Rössler systems. It was shown that the averaging method is successful for a certain type of zero-Hopf critical points, but fails for some type of such critical points. In this paper, we apply normal form theory to reinvestigate the bifurcation and show that the method of normal forms is applicable for all types of zero-Hopf bifurcations, revealing why the time-averaging method fails for some type of zero-Hopf bifurcation.

中文翻译：

---

使用范式理论分析两个罗斯勒系统中的零 Hopf 分岔

在最近的出版物中 [Llibre, 2014; Llibre & Makhlouf, 2020]，时间平均方法被应用于研究从两个罗斯勒系统的零 Hopf 临界点分叉的周期轨道。结果表明，平均方法对于某种类型的零 Hopf 临界点是成功的，但对于某种类型的这种临界点是失败的。在本文中，我们应用范式理论来重新研究分岔，并表明范式方法适用于所有类型的零 Hopf 分岔，揭示了为什么时间平均方法对于某些类型的零 Hopf 分岔失败。