Phase models and clustering in networks of oscillators with delayed, all-to-all coupling

We consider a general model for a network of all-to-all coupled oscillators with time delayed connections. We reduce the system of delay differential equations to a phase model where the time delay enters as a phase shift. By analyzing the phase model, we study the existence and stability of cluster solutions. These are solutions where the oscillators divide into groups; oscillators within a group are synchronized, while oscillators in different groups are phase-locked with a fixed phase difference. We show that the time delay can lead to the multistability between different cluster states. Analytical results are compared with numerical studies of the full system of delay differential equations.