在四维空间中异宿临界网络附近动力学的分析：投影映射
Analysis of dynamics near heteroclinic networks in $\mathbb{R}^{4}$ with a projected map
AI总结：
本文研究四维空间中三个异宿临界网络附近轨迹的动力学，引入投影映射方法分析稳定性，发现轨迹切换可由折叠分岔或边界碰撞分岔引起，并解决了一个30年的猜想。
摘要：
异宿环和网络是动力系统中的结构，由不变集和连接异宿轨道组成，可以在具有不变子空间的系统中保持鲁棒性。通常分析异宿环和网络稳定性的方法是构造返回映射到网络附近的横截面。从这些返回映射中可以定义转移矩阵，其特征值和特征向量可用于确定稳定性。本文引入了一种扩展方法，即投影映射，通过将轨迹与在某种意义上具有相同动力学的轨迹识别，定义为离散的、分段光滑的一维映射，其维度比转移矩阵的秩少一。我们使用这些映射来描述四个平衡点的三个异宿临界网络附近轨迹的动力学。我们发现，在所有三种情况下，轨迹在环网络之间切换的起始可以由折叠分岔或边界碰撞分岔引起，其中映射的固定点不再存在于相应函数的定义域中。我们能够证明，这些三个网络附近的轨迹仅渐近于一个子环，不能多次切换子环，与具有更多平衡点的例子不同，并解决了Brannath提出的30年-old猜想。我们还能够将某些结果推广到所有准简单网络，证明投影映射的边界碰撞分岔，对应于某些转移矩阵特征向量的条件，会导致环失去稳定性。
英文摘要：
Heteroclinic cycles and networks are structures in dynamical systems composed of invariant sets and connecting heteroclinic orbits, and can be robust in systems with invariant subspaces. The usual method for analysing the stability of heteroclinic cycles and networks is to construct return maps to cross-sections near the network. From these return maps, transition matrices can be defined, whose eigenvalues and eigenvectors can be analysed to determine stability. In this paper, we introduce an extension to this methodology, the projected map, which we define by identifying trajectories with, in a certain sense, qualitatively the same dynamics. The projected map is a discrete, piecewise-smooth map of one dimension one fewer than the rank of the transition matrix. We use these maps to describe the dynamics of trajectories near three heteroclinic networks in $\mathbb{R}^{4}$ with four equilibria. We find in all three cases that the onset of trajectories that switch between cycles of the network can be caused by a fold bifurcation or border-collision bifurcation, where fixed points of the map no longer exist in the corresponding function's domain of definition. We are able to show that trajectories near these three networks are asymptotic to only one subcycle, and cannot switch between subcycles multiple times, in contrast to examples with more than four equilibria, and resolving a 30-year-old claim by Brannath. We are also able to generalise some results to all quasi-simple networks, showing that a border-collision bifurcation of the projected map, corresponding to a condition on the eigenvectors of certain transition matrices, causes a cycle to lose stability.
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