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

Abstract

We examine the dynamics of an ensemble of phase oscillators that are divided in k sets, with timedelayed coupling interactions only between oscillators in different sets or partitions. The network of interactions thus form a k−partite graph. We observe a variety of phase-locked states that include, in addition to the in-phase fully synchronized solution, a variety of splay cluster solutions; all oscillators within a partition are synchronised and the phase differences between oscillators in different partitions are 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 for the k−partite system. With increase in time-delay, there is an increase in multistability: the above generic solutions coexist with a number of other partially synchronized solutions. We apply the Ott-Antonsen ansatz for the special case of a symmetric k−partite graph to obtain a single time-delayed differential equation for the attracting synchronization manifold. The agreement with numerical results for the specific case of oscillators on a tripartite lattice (the k = 3 case) is excellent.