Stability and bifurcation for the Kuramoto model

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• 作者：Helge Dietert¹ ; ^{H.G.W.Dietert@maths.cam.ac.uk" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35Q83 ; 35Q92 ; 35B32 ; 35B40 ; 82C27
• 刊名：Journal de mathematiques pures et appliquees
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：105
• 期：4
• 页码：451-489
• 全文大小：669 K

文摘

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,1${\mathcal{W}}^{n,1}$ for 11057" title="Click to view the MathML source">n>1$n>1$. 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.