发表机构
Universitat Autònoma de Barcelona; Reial Acadèmia de Ciències i Arts de Barcelona; University of Zielona Góra（巴塞罗那自治大学; 巴塞罗那皇家科学与艺术学院; 谢德涅戈拉大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究经典Rössler系统的零-Hopf分岔，证明在适当参数条件下有三族周期轨道从零-Hopf平衡点分岔，并利用平均理论与Poincaré准则证明这些周期轨道构成$C^{1}$局部一阶积分存在的通用障碍。
AI 中文摘要
我们研究了原始Rössler系统的零-Hopf分岔和局部$C^{1}$不可积性。首先，我们刻画了该系统唯一的零-Hopf平衡点，并利用一阶平均理论研究了其扰动。我们证明，在关于分岔参数的适当假设下，三族不同的周期轨道从零-Hopf平衡点分岔出来，这大大扩展了先前已知的仅获得一个分岔周期轨道的结果。在论文的第二部分，这些分岔周期解被用来研究Rössler系统的可积性性质。通过将一阶平均理论与经典的一阶积分Poincaré准则相结合，我们证明了分岔周期轨道构成了类$C^{1}$局部一阶积分存在的通用障碍。这表明平均理论不仅为检测周期解提供了有效工具，而且为可积性分析提供了一种强有力的方法。
英文摘要
We investigate the zero-Hopf bifurcation and the local $C^{1}$-non-integrability of the original Rössler system. First, we characterize the unique zero-Hopf equilibrium of the system and study its perturbation by means of the first-order averaging theory. We prove that, under suitable assumptions on the bifurcation parameters, three distinct families of periodic orbits bifurcate from the zero-Hopf equilibrium, substantially extending the previously known result in which only one bifurcating periodic orbit was obtained. In the second part of the paper, these bifurcating periodic solutions are used to investigate the integrability properties of the Rössler system. By combining the first-order averaging theory with the classical Poincaré criterion for first integrals, we prove that the bifurcating periodic orbits constitute a generic obstruction to the existence of local first integrals of class $C^{1}$. This demonstrates that averaging theory provides not only an effective tool for detecting periodic solutions but also a powerful method for the integrability analysis.
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