AI 中文总结
该研究基于不变球面定理，在环耦合系统中推导全局吸引不变球面及异宿网络的存在条件，分析扰动下异宿网络的破坏与周期轨道的出现，利用对称性简化异宿网络并构建几何分析框架。
AI 中文摘要
在本研究中，我们展示了异宿网络如何通过应用不变球面定理在一类简单的网络动力系统中产生。环耦合系统是定义在ℝⁿ上的常微分方程网络，其中每个变量xᵢ仅与其前一个变量xᵢ₋₁相互作用。我们推导了三次多项式系数满足的条件，这些条件通过不变球面定理保证全局吸引不变球面的存在性。此外，将其中一个系数设为零，我们确定了剩余系数满足的条件，这些条件保证不变球面上异宿网络的存在性。针对n=3的情况，我们研究了异宿网络邻域内消失系数的扰动。在这类扰动下，异宿网络被破坏，周期轨道出现。当原始异宿网络渐近稳定时，产生的周期轨道会追踪该网络；在一个参数范围内，会出现唯一的吸引周期轨道并追踪整个异宿网络。相比之下，当异宿网络不稳定时，会出现排斥周期轨道，它们仅追踪异宿结构的一部分，即一半的异宿连接。对称性在整个分析中起着关键作用。利用系统的对称性，我们将异宿网络简化为两个同宿轨道。此外，异宿网络附近的局部动力学可在由环带连接莫比乌斯带构成的商空间上研究，每个同宿轨道分别位于这两个分量上。这种简化为理解分岔和周期动力学的产生提供了几何框架。
英文摘要
In this work, we show how heteroclinic networks can arise in a simple class of network dynamical systems through the application of the Invariant Sphere Theorem. Ring-coupled systems are ODE networks in $\mathbb{R}^n$ in which each variable $x_i$ interacts only with its predecessor $x_{i-1}$. We derive conditions on the coefficients of a cubic polynomial that guarantee the existence of a globally attracting invariant sphere via the Invariant Sphere Theorem. Moreover, setting one of these coefficients to zero, we identify conditions on the remaining coefficients that guarantee the existence of a heteroclinic network on the invariant sphere. Focusing on the case $n=3$, we investigate perturbations of the vanishing coefficient in a neighbourhood of the heteroclinic network. Under such perturbations, the heteroclinic network is destroyed and periodic orbits emerge. When the original heteroclinic network is asymptotically stable, the resulting periodic orbits shadow the network. In one parameter regime, a unique attracting periodic orbit appears and shadows the entire heteroclinic network. By contrast, when the heteroclinic network is not stable, repelling periodic orbits arise that shadow only part of the heteroclinic structure, namely half of the heteroclinic connections. Symmetry plays a fundamental role throughout the analysis. Exploiting the symmetries of the system, we reduce the heteroclinic network to two homoclinic orbits. Furthermore, the local dynamics near the heteroclinic network can be studied on a quotient space consisting of an annulus attached to a Möbius band, with each homoclinic orbit lying on one of these components. This reduction provides a geometric framework for understanding the bifurcations and the emergence of the periodic dynamics.
