弱耦合振子的同步：耦合、时滞和拓扑
Synchronization of Weakly Coupled Oscillators: Coupling, Delay and Topology
摘要：
耦合振子系统有三个关键因素来表征它们之间的相互作用：耦合（如何影响），延迟（何时影响）和拓扑（影响谁）。对于其中每一个，现有的工作主要集中在特殊情况下。本文从新的角度和工具出发，放宽了对这些因素的某些假设。本文主要有三个结果。首先，利用代数图论的结果，得到了一个判定平衡点稳定性的充分条件。此条件适用于任意拓扑。它推广了已有的结果，也导致了一个充分条件的耦合功能，该系统是保证达到同步。其次，已知具有sin（）耦合函数的相同振荡器保证在完全图上相位同步。使用我们的结果，我们表明，在许多情况下，某些结构，而不是确切的形状的耦合功能，如对称性和对称性是全球同步的关键。最后，研究了非均匀时滞的影响。我们开发了一个新的框架，通过构建一个非延迟相位模型，近似于原来的连续极限。我们进一步推导出它的稳定性如何依赖于时滞分布。特别是，我们表明，异质性，即更广泛的延迟分布，可以帮助达到同相同步。
英文摘要：
There are three key factors of a system of coupled oscillators that characterize the interaction among them: coupling (how to affect), delay (when to affect) and topology (whom to affect). For each of them, the existing work has mainly focused on special cases. With new angles and tools, this paper makes progress in relaxing some assumptions of these factors. There are three main results in this paper. First, by using results from algebraic graph theory, a sufficient condition is obtained which can be used to check equilibrium stability. This condition works for arbitrary topology. It generalizes existing results and also leads to a sufficient condition on the coupling function with which the system is guaranteed to reach synchronization. Second, it is known that identical oscillators with sin() coupling functions are guaranteed to synchronize in phase on a complete graph. Using our results, we demonstrate that for many cases certain structures instead of exact shape of the coupling function such as symmetry and concavity are the keys for global synchronization. Finally, the effect of heterogenous delays is investigated. We develop a new framework by constructing a non-delayed phase model that approximates the original one in the continuum limit. We further derive how its stability properties depend on the delay distribution. In particular, we show that heterogeneity, i.e. wider delay distribution, can help reach in-phase synchronization.
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