| The phenomenon of frequency synchronization is ubiquitous and is observed when the individual frequencies of oscillators converge to a common value via coupling. A celebrated model for the synchronization phenomena of oscillators is due to Kuramoto. A lot of research about synchronization has been carried out based on the Kuramoto model [1,20,27]. The Kuramoto model has been studied in biology, physics, chemistry, sociology and so on. We discuss the asymptotic complete frequency synchronization for locally interacting Kuramoto oscillators instead of all-to-all interaction. In this paper we study the influence of the inertial effect on frequency synchronization in an ensemble of Kuramoto oscillators with finite inertia and symmetric and connected interactions. We present sufficient conditions in terms of coupling strength, algebraic connectivity, natural frequencies, and the inertial coefficient to guarantee the occurrence of frequency synchronization. During the process, we apply the LaSalle’s invariance principle and Laplacian matrix to simplify the frequency synchronization problem. We also make a comparison with the existing conditions proposed for the first-order Kuramoto model and conclude that the inertial effect, if appropriately small, has little influence as far as frequency synchronization is concerned. |
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