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

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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.