Asymptotic behavior of the Kuramoto system with periodic natural frequency

We study the asymptotic behavior of the Kuramoto system with time-periodic natural frequencies and all-to-all coupling. For a relatively large coupling strength K>0 compared to the differences between the time-periodic natural frequencies, we obtain the existence of time-periodic states variables θ_ij=θ_i−θ_j. We derive an explicit formula for the total phase of the solution, which depends on the initial data and natural frequencies. We prove that if the solutions to the Kuramoto system with all-to-all coupling have the same total phase, then their phase differences θ_ij for any corresponding initial data converge to the same periodic state θ_ij^T as t goes to ∞. Finally, by combining the total phase formula and this stability, we present a sharp asymptotic formula consisting of the time-periodic state, drift term, and exponentially decaying error term. The drift term depends only on the natural frequency and initial data. Several simulations are carried out to validate the proposed results.