Asymptotic analysis of the time-periodic Kuramoto system on general symmetric coupling and phase shift structures

We consider the asymptotic behavior of the solution to the time-periodic Kuramoto system with general symmetric coupling and phase shifts. To cover all symmetric couplings, we employ a new framework that consists of the maximal diameter of phase difference estimates and the ℓ2-upper bound of mean-centered variables. Using this framework, we show the existence of a time-periodic solution to the corresponding mean-centered model for time-periodic coefficients. We also provide a sufficient condition for stability: the phase differences of the two solutions to the time-periodic Kuramoto system converge to the same state despite phase shifts and lack of all-to-all coupling. Combining these properties, we prove that the phase difference of the time-periodic Kuramoto model with general symmetric coupling and phase shifts converges exponentially to the time-periodic state. We present some numerical simulations to validate our theoretical results.