论文标题
偶联振荡器环中杂斜周期的稳定性
Stability of heteroclinic cycles in rings of coupled oscillators
论文作者
论文摘要
通过边缘连接的相互作用节点的网络在科学询问的几乎每个分支中都出现。网络的连通性结构可以迫使不变子空间的存在，这在通用动力系统中不会出现。这些不变的子空间可能会导致出现可靠的杂斜周期，否则在结构上将是不稳定的。通常，稳定的杂斜周期附近的动力学是非连接的：循环中固定点附近的平均停留时间是不确定的，并且存在持续的放缓。在本文中，我们检查了用最近的邻次或最近的m-邻近耦合的环网络，并证明在动力学的相位空间中存在杂斜周期的类别。我们表明，总是至少有一个杂斜周期，这可能是渐近稳定的，因此，网络的吸引动力学预计将是非共性的。我们推测，这种行为的大部分持续在结构较低的网络中，因此，非共性行为是典型的。
Networks of interacting nodes connected by edges arise in almost every branch of scientific enquiry. The connectivity structure of the network can force the existence of invariant subspaces, which would not arise in generic dynamical systems. These invariant subspaces can result in the appearance of robust heteroclinic cycles, which would otherwise be structurally unstable. Typically, the dynamics near a stable heteroclinic cycle is non-ergodic: mean residence times near the fixed points in the cycle are undefined, and there is a persistent slowing down. In this paper, we examine ring networks with nearest-neighbour or nearest-m-neighbour coupling, and show that there exist classes of heteroclinic cycles in the phase space of the dynamics. We show that there is always at least one heteroclinic cycle which can be asymptotically stable, and thus the attracting dynamics of the network are expected to be non-ergodic. We conjecture that much of this behaviour persists in less structured networks and as such, non-ergodic behaviour is somehow typical.
