Synchronization Conditions of a Mixed Kuramoto Ensemble in Attractive and Repulsive Couplings

We study the emergent dynamics of a mixed Kuramoto ensemble in the presence of both attractive and repulsive coupling strengths. To be precise, we consider coupled Kuramoto-type systems consisting of two ensembles in which the oscillators in the same group interact attractively with a positive intra-group coupling strength, whereas the oscillators in the different group communicate repulsively with a negative inter-group coupling strength. For the modeling perspective of the Kuramoto model (KM for brevity), two types of systems are treated in this paper: an attractive-repulsive inertial KM-KM and an attractive-repulsive KM-KM. For these two models, we provide sufficient frameworks leading to complete bi-polar synchronization in which asymptotic configuration tends to a bi-polar state. Our estimates mainly rely on the gradient-like flow formulation and the energy estimate.

Automated summary

This study examines the emergent dynamics of a mixed Kuramoto ensemble with attractive and repulsive coupling strengths, finding that complete bi-polar synchronization can be achieved when considering coupling between two ensembles. The research relies on gradient-like flow formulation and energy estimate for the modeling perspective of the Kuramoto model.