Emergence of Synchronization in Coupled Dynamical Systems:Influence of Network Topology and Coupling Delay

同步现象在自然、科学、工程和社会学中普遍存在。不同单位寻求同步运作的趋势吸引了大量的研究兴趣。在其他因素中，网络拓扑和时延起着重要的作用。耦合细胞网络的形式化给出了用耦合常微分方程及其相关分支系统研究同步问题的一般框架。等变度理论为研究不动点、周期轨道及其分支的基本性质提供了一种拓扑工具。本提案的主要目标是利用耦合细胞网络和等变度理论的思想，研究、分析和测量网络拓扑结构和耦合延迟对同步过程的影响。首先，我们想开发一种拓扑方法来研究耦合细胞网络中与同步相关的分支。其次，我们希望将其与现有的耦合单元网络方法相结合，以更深入地了解网络拓扑的影响。第三，利用耦合延迟微分方程和耦合细胞网络的组合方法研究耦合延迟的影响。最后，我们要继续研究等变分岔理论中的Ize猜想，该猜想指出任何具有绝对不可约群作用的分岔问题总是导致稳态分岔。

Synchronization phenomena are ubiquitous in nature, science, engineering and sociology. The tendency of different units seeking to operate in synchrony has attracted a large amount of research interest. Among other factors, network topology and time delay are found to play an important role. Formalism of coupled cell networks gives a general framework for studying synchronization using systems of coupled ordinary differential equations and their associated bifurcations. Equivariant degree theory provides a topological tool in studying essential properties of fixed points, periodic orbits and their related bifurcations. The main goal of this proposal is to investigate, analyze and measure the influence of network topology and coupling delay upon the synchronization process, using ideas from coupled cell networks and equivariant degree theory. At first, we want to develop a topological method to study synchronization related bifurcations in coupled cell networks. Secondly, we want to combine it with existing methods of coupled cell networks, to gain a deeper insight on impact of network topology. Thirdly, we want to study the influence of coupling delay using coupled delay differential equations and the combined method used for coupled cell networks. Last but not least, we want to continue to work on the Ize conjecture in equivariant bifurcation theory, which states that any bifurcation problem with absolutely irreducible group actions always leads to bifurcation of steady states.