Complete synchronization of Kuramoto oscillators with finite inertia

We present an approach based on Gronwall’s inequalities for the asymptotic complete phase–frequency synchronization of Kuramoto oscillators with finite inertia. For given finite inertia and coupling strength, we present admissible classes of initial configurations and natural frequency distributions, which lead to the complete phase–frequency synchronization asymptotically. For this, we explicitly identify invariant regions for the Kuramoto flow, and derive second-order Gronwall’s inequalities for the evolution of phase and frequency diameters. Our detailed time-decay estimates for phase and frequency diameters are independent of the number of oscillators. We also compare our analytical results with numerical simulations.