On the Critical Coupling for Kuramoto Oscillators

The celebrated Kuramoto model captures various synchronization phenomena in biological and man-made dynamical systems of coupled oscillators. It is well-known that there exists a critical coupling strength among the oscillators at which a phase transition from incoherency to synchronization occurs. This paper features four contributions. First, we characterize and distinguish the different notions of synchronization used throughout the literature and formally introduce the concept of phase cohesiveness as an analysis tool and performance index for synchronization. Second, we review the vast literature providing necessary, sufficient, implicit, and explicit estimates of the critical coupling strength in the finite and infinite-dimensional case and for both first-order and second-order Kuramoto models. Third, we present the first explicit necessary and sufficient condition on the critical coupling strength to achieve synchronization in the finite-dimensional Kuramoto model for an arbitrary distribution of the natural frequencies. The multiplicative gap in the synchronization condition yields a practical stability result determining the admissible initial and the guaranteed ultimate phase cohesiveness as well as the guaranteed asymptotic magnitude of the order parameter. As supplementary results, we provide a statistical comparison of our synchronization condition with other conditions proposed in the literature, and we show that our results also hold for switching and smoothly time-varying natural frequencies. Fourth and finally, we extend our analysis to multi-rate Kuramoto models consisting of second-order Kuramoto oscillators with inertia and viscous damping together with first-order Kuramoto oscillators with multiple time constants. We prove that such a heterogeneous network is locally topologically conjugate to a first-order Kuramoto model with scaled natural frequencies. Finally, we present necessary and sufficient conditions for almost global phase synchronization and local frequency synchronization in the multi-rate Kuramoto model. Interestingly, our provably correct synchronization conditions do not depend on the inertiae which contradicts prior observations on the role of inertial effects in synchronization of second-order Kuramoto oscillators.