具有塑性耦合的相位耦合振荡器：同步与稳定性
Phase-coupled Oscillators with Plastic Coupling: Synchronization and Stability
摘要：
本文研究了具有塑性耦合强度和任意拓扑结构的均匀相位耦合振子系统的同步问题。两个振荡器之间的耦合强度的动态由这些振荡器之间的相位差决定。我们证明了，在温和的假设下，这样的系统是梯度系统，并且总是实现频率同步。此外，我们提供了充分的稳定性和不稳定性的条件是基于代数图论的结果。对于底层拓扑为树的特殊情况，我们给出了平衡点稳定的一个判据（充要条件）。对于树拓扑和任意拓扑，我们提供了锁相的充分条件，即几乎必然收敛到稳定平衡点。另外，我们还找到了系统具有唯一稳定平衡点的条件，因此，几乎全局稳定性如下。几个例子被用来证明各种平衡的系统，它们对系统的参数的依赖，并说明在行为的系统与恒定和塑性耦合强度的差异。
英文摘要：
In this article we study synchronization of systems of homogeneous phase-coupled oscillators with plastic coupling strengths and arbitrary underlying topology. The dynamics of the coupling strength between two oscillators is governed by the phase difference between these oscillators. We show that, under mild assumptions, such systems are gradient systems, and always achieve frequency synchronization. Furthermore, we provide sufficient stability and instability conditions that are based on results from algebraic graph theory. For a special case when underlying topology is a tree, we formulate a criterion (necessary and sufficient condition) of stability of equilibria. For both, tree and arbitrary topologies, we provide sufficient conditions for phase-locking, i.e. convergence to a stable equilibrium almost surely. We additionally find conditions when the system possesses a unique stable equilibrium, and thus, almost global stability follows. Several examples are used to demonstrate variety of equilibria the system has, their dependence on system's parameters, and to illustrate differences in behavior of systems with constant and plastic coupling strengths.
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