Synchronization of weakly coupled oscillators: coupling, delay and topology

There are three key factors in a system of coupled oscillators that characterize the interaction between them: coupling (how to affect), delay (when to affect) and topology (whom to affect). The existing work on each of these factors has mainly focused on special cases. With new angles and tools, this paper makes progress in relaxing some assumptions on these factors. There are three main results in this paper. Firstly, by using results from algebraic graph theory, a sufficient condition is obtained that can be used to check equilibrium stability. This condition works for arbitrary topology, generalizing existing results and also leading to a sufficient condition on the coupling function which guarantees that the system will reach synchronization. Secondly, it is known that identical oscillators with sin() coupling functions are guaranteed to synchronize in phase on a complete graph. Our results prove that in many cases certain structures such as symmetry and concavity, rather than the exact shape of the coupling function, are the keys for global synchronization. Finally, the effect of heterogenous delays is investigated. Using mean field theory, a system of delayed coupled oscillators is approximated by a non-delayed one whose coupling depends on the delay distribution. This shows how the stability properties of the system depend on the delay distribution and allows us to predict its behavior. In particular, we show that for sin() coupling, heterogeneous delays are equivalent to homogeneous delays. Furthermore, we can use our novel sufficient instability condition to show that heterogeneity, i.e. wider delay distribution, can help reach in-phase synchronization.