On the Critical Coupling of the Finite Kuramoto Model on Dense Networks

Kuramoto model is one of the most prominent models for the synchronization of coupled oscillators. It has long been a research hotspot to understand how natural frequencies, the interaction between oscillators, and network topology determine the onset of synchronization. In this paper, we investigate the critical coupling of Kuramoto oscillators on deterministic dense networks, viewed as a natural generalization from all-to-all networks of identical oscillators. We provide a sufficient condition under which the Kuramoto model with non-identical oscillators has one unique and stable equilibrium. Moreover, this equilibrium is phase cohesive and enjoys local exponential synchronization. We perform numerical simulations of the Kuramoto model on random networks and circulant networks to complement our theoretical analysis and provide insights for future research.