On mathematical analysis of synchronization of bidirectionally coupled Kuramoto oscillators under inertia effect

In this article, we analyze the phase synchronization and frequency synchronization for the bidirectionally coupled Kuramoto model under the effect of inertia. Unlike the classical Kuramoto model equipped with all-to-all coupled interaction, in the setting of this model, each oscillator ${\theta }_i$ only interacts directly with ${\theta }_{i+1}$ and ${\theta }_{i-1}$. The bidirectional interaction is a typical setting of the concatenation in power systems. Additionally, it is necessary to impose the effect of inertia in the Kuramoto model in the applications such as power systems and Josephson junction array. In this article, we first present a theory of the global convergence for frequency synchronization for the identical case. For the non-identical case, we prove that the second-order bidirectionally coupled Kuramoto model exhibits a frequency synchronization if the coupling strength is large, inertia is small, and all oscillators are initially confined to a sector. We emphasize that the arc length of this sector possesses a positive lower bound which is independent of the number of oscillators. If, in addition, all natural frequencies are identical, we further show that the phase synchronization emerges. Moreover, we demonstrate the numerical simulations to support the main results. On the other hand, we observe that the model equipped with large inertia can exhibit the synchronization. Exploring the synchronization theory for large inertia case is left as the future work.