Stability of Twisted States in the Continuum Kuramoto Model

We study a nonlocal diffusion equation approximating the dynamics of coupled phase oscillators on large graphs. Under appropriate assumptions, the model has a family of steady state solutions called twisted states. We prove a sufficient condition for stability of twisted states with respect to perturbations in the Sobolev and BV spaces. As an application, we study the stability of twisted states in the Kuramoto model on small-world graphs.