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

Stability of twisted states in the continuum Kuramoto model