Stability and bifurcation for the Kuramoto model

Abstract

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 ${\mathcal{W}}^{n,1}$ 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.

33 pages, 2 figures

References in corpus (1)

• Exact Results for the Kuramoto Model with a Bimodal Frequency Distribution

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