Synchronization of Kuramoto oscillators with the distributed time-delays and inertia effect

We prove the complete and partial phase synchronization for the Kuramoto oscillators with distributed time-delays and inertia effect. Our results assert that the Kuramoto models incorporated with a small variation of distributed time-delays and inertia effect still exhibit synchronization. This shows the robustness of the synchronization phenomena of the original Kuramoto model in the perturbation of time-delay and inertia effects. We also present several numerical experiments supporting our main results and exhibiting interesting patterns.