Kuramoto模型的不稳定流形：收敛至Ott-Antonsen流形
Unstable Manifolds for the Kuramoto Model: Convergence to the Ott-Antonsen Manifold
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中文总结 AI 辅助
本文对全耦合Kuramoto模型的平衡点做谱分析，推导非相干平衡点的不稳定流形，证明其收敛至Ott-Antonsen流形，还分析连续极限下的对应情况，建立有限维粒子系统与连续极限的几何关联。
中文摘要 AI 辅助
本文研究有限维、均匀全耦合的Kuramoto模型。首先对系统所有平衡点进行完整谱分析，受该分析启发，推导非相干平衡点族对应的不稳定流形的显式描述；随后，基于p- Wasserstein度量诱导的Hausdorff距离，确立该不稳定流形族收敛至Ott-Antonsen流形$\boldsymbol{\textit{M}}_{\boldsymbol{\text{OA}}}$；进一步对连续极限下$\boldsymbol{\textit{M}}_{\boldsymbol{\text{OA}}}$的对应对象开展类似分析，研究结果为有限维粒子系统与其平均场或连续极限之间提供了直接几何关联。
英文摘要
In this paper, we study the finite-dimensional, homogeneous, all-to-all coupled Kuramoto model. We begin by performing a complete spectral analysis of all equilibria of the system. Motivated by this analysis, we then derive an explicit description of the unstable manifolds associated with the family of incoherent equilibria. Subsequently, we establish the convergence of this family of unstable manifolds to the Ott-Antonsen manifold $\mathcal{M}_{\mathrm{OA}}$, with respect to the Hausdorff distance induced by the $p$-Wasserstein metric. We further carry out an analogous analysis for the corresponding counterpart of $\mathcal{M}_{\mathrm{OA}}$ in the continuum limit. Moreover, we establish the uniform-in-time convergence of trajectories of the finite-dimensional Kuramoto model on these invariant manifolds towards their corresponding mean-field limit trajectories. Our results provide a direct geometric link between finite-dimensional particle systems and their mean-field, or continuum, limits.
发表机构
- Technical University of Munich, School of Computation, Information and Technology（慕尼黑工业大学计算、信息和技术学院）
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