Exponential Asymptotic Stability of the Kuramoto System with Periodic Natural Frequencies and Constant Inertia

We consider the temporal asymptotic behavior of the all-to-all coupled Kuramoto model with inertia and time-periodic natural frequencies. Due to the inertial effect, there are three cases of the dynamical ensemble with respect to the coupling strength; large coupling, near boundary, and small coupling. For each case, we present the asymptotic behavior of the solution to the inertial Kuramoto model with periodic natural frequencies: the solutions commonly consist of a macroscopic phase, a mean-centered-periodic solution, and an exponential decay term. The macroscopic phase is a drift-type term determined by initial data and natural frequencies, and the mean-centered-periodic solution is a standing wave independent of initial data. We provide sufficient conditions for the existence of a mean-centered-periodic solution with a time-periodic phase difference between nodes for each case and its exponential stability. We also provide several simulations to confirm our mathematical results.