Stability and bifurcation for the Kuramoto model
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arXiv
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| Format: | Preprint |
| Publié: |
2014
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| _version_ | 1866915209472376832 |
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| author | Dietert, Helge |
| author_facet | Dietert, Helge |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1411_3752 |
| institution | arXiv |
| publishDate | 2014 |
| record_format | arxiv |
| spellingShingle | Stability and bifurcation for the Kuramoto model Dietert, Helge Analysis of PDEs Dynamical Systems 35Q83, 35Q92, 35B32, 35B40, 82C27, 92B25, 37N25 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. |
| title | Stability and bifurcation for the Kuramoto model |
| topic | Analysis of PDEs Dynamical Systems 35Q83, 35Q92, 35B32, 35B40, 82C27, 92B25, 37N25 |
| url | https://arxiv.org/abs/1411.3752 |