Phase-locking in k-partite networks of delay-coupled oscillators

We examine the dynamics of an ensemble of phase oscillators that are divided in k sets, with time-delayed coupling interactions only between oscillators in different sets or partitions. The network of interactions thus forms a k−partite graph. A variety of phase-locked states are observed; these include, in addition to the fully synchronized in-phase solution, splay cluster solutions in which all oscillators within a partition are synchronised and the phase differences between oscillators in different partitions are integer multiples of 2π/k. Such solutions exist independent of the delay and we determine the generalised stability criteria for the existence of these phase-locked solutions. With increase in time-delay, there is an increase in multistability, the generic solutions coexisting with a number of other partially synchronized solutions. The Ott-Antonsen ansatz is applied for the special case of a symmetric k−partite graph to obtain a single time-delayed differential equation for the attracting synchronization manifold. Agreement with numerical results for the specific case of oscillators on a tripartite lattice (the k=3 case) is excellent.