| Popis: |
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 i j = 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 i j for any corresponding initial data converge to the same periodic state θ i j 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. |
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