发表机构
Instituto de Ciências Puras e Aplicadas, Universidade Federal de Itajubá（阿皮卡斯科学学院，伊塔比乌巴联邦大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文利用二阶平均理论研究了Hopf-Langford型系统中零-Hopf平衡点分岔出的周期解与不变环面，证明环面不稳定，并给出数值例证。
AI 中文摘要
我们研究了在Hopf-Langford型系统中从零-Hopf平衡点分岔出的周期解和不变环面的存在性。利用二阶平均理论，我们建立了显式的参数条件，在这些条件下系统在原点附近存在一个周期解，并给出了其稳定性。随后，我们应用最近关于不变环面与相关Poincaré映射的Neimark-Sacker分岔之间关系的结果，得到一条光滑的分岔曲线，沿着该曲线存在一个唯一的不变环面包围着该周期解。相关的第一Lyapunov系数被显式地计算为扰动参数的幂级数，并且我们证明在所考虑的条件下，该系数始终为正；因此，只要该环面存在，无论在我们假设范围内选择何种具体参数值，分岔出的环面都是不稳定的。这为该系统族中渐近稳定环面的存在提供了一个结构性障碍。我们通过两个数值例子说明了我们的结果，一个例子展示了孤立的渐近稳定周期解且没有分岔环面，另一个例子展示了被不稳定不变环面包围的周期解，后者也通过其Poincaré映射进行了说明。
英文摘要
We study the existence of periodic solutions and invariant tori bifurcating from a zero-Hopf equilibrium in a Hopf-Langford type system. Using second-order averaging theory, we establish explicit parameter conditions under which the system admits a periodic solution near the origin, together with its stability. We then apply recent results relating invariant tori to Neimark-Sacker bifurcations of the associated Poincaré map to obtain a smooth bifurcation curve along which a unique invariant torus surrounds the periodic solution. The relevant first Lyapunov coefficient is computed explicitly as a power series in the perturbation parameter, and we show that, under the conditions considered, this coefficient is always positive; consequently, whenever it exists, the bifurcating torus is unstable, regardless of the specific parameter values chosen within our hypotheses. This provides a structural obstruction to the existence of asymptotically stable tori in this family of systems. We illustrate our results with two numerical examples, one exhibiting an isolated asymptotically stable periodic solution with no bifurcating torus, and another exhibiting a periodic solution surrounded by an unstable invariant torus, the latter also illustrated through its Poincaré map.
