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Synchronization of second-order inhomogeneous Kuramoto oscillator systems with time delays and phase shifts

In this paper, a second-order inhomogeneous Kuramoto oscillator systems(ihKOSs) with communication delays and phase shifts are investigated. An equivalent form of the second-order ihKOSs is obtained by using singular perturbation analysis. Both of phase synchronization and frequency synchronization are studied under an undirected connected network topology. Some criterion for the phase and frequency synchronization of the second-order ihKOSs are obtained. Finally, the validity and correctness of the results are further verified by some numerical simulations.