Stability and bifurcation for the Kuramoto model

We study the mean-field limit of the Kuramoto model of globally coupled oscillators. By studying the evolution in Fourier space and understanding the domain of dependence, we show a global stability result. Moreover, we can identify function norms to show damping of the order parameter for velocity distributions and perturbations in W-n,W-1 for n > 1. Finally, for sufficiently regular velocity distributions we can identify exponential decay in the stable case and otherwise identify finitely many eigenmodes. For these eigenmodes we can show a center unstable manifold reduction, which gives a rigorous tool to obtain the bifurcation behaviour. The damping is similar to Landau damping for the Vlasov equation. (C) 2015 The Author. Published by Elsevier Masson SAS.