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

TL;DR

This paper studies the dynamics of delay-coupled phase oscillators arranged in a k-partite network, revealing various stable phase-locked states, their stability criteria, and the effects of delay on multistability, with analytical and numerical validation.

Contribution

It introduces a comprehensive analysis of phase-locked states in k-partite networks with delay, including stability conditions and the application of Ott-Antonsen ansatz for symmetric cases.

Findings

Existence of multiple phase-locked solutions independent of delay

Delay increases multistability and coexistence of solutions

Analytical results agree with numerical simulations for tripartite case

Abstract

We examine the dynamics of an ensemble of phase oscillators that are divided in $k$ sets, with time-delayed coupling interactions {\em 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 $2\pi/k$. 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.