The Kuramoto model on a sphere: Explaining its low-dimensional dynamics with group theory and hyperbolic geometry
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| Format: | Preprint |
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2019
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| author | Lipton, Max Mirollo, Renato Strogatz, Steven H. |
| author_facet | Lipton, Max Mirollo, Renato Strogatz, Steven H. |
| contents | We study a system of $N$ interacting particles moving on the unit sphere in $d$-dimensional space. The particles are self-propelled and coupled all to all, and their motion is heavily overdamped. For $d=2$, the system reduces to the classic Kuramoto model of coupled oscillators; for $d=3$, it has been proposed to describe the orientation dynamics of swarms of drones or other entities moving about in three-dimensional space. Here we use group theory to explain the recent discovery that the model shows low-dimensional dynamics for all $N \ge 3$, and to clarify why it admits the analog of the Ott-Antonsen ansatz in the continuum limit $N \rightarrow \infty$. The underlying reason is that the system is intimately connected to the natural hyperbolic geometry on the unit ball $B^d$. In this geometry, the isometries form a Lie group consisting of higher-dimensional generalizations of the Möbius transformations used in complex analysis. Once these connections are realized, the reduced dynamics and the generalized Ott-Antonsen ansatz follow immediately. This framework also reveals the seamless connection between the finite and infinite-$N$ cases. Finally, we show that special forms of coupling yield gradient dynamics with respect to the hyperbolic metric, and use that fact to obtain global stability results about convergence to the synchronized state. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1907_07150 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | The Kuramoto model on a sphere: Explaining its low-dimensional dynamics with group theory and hyperbolic geometry Lipton, Max Mirollo, Renato Strogatz, Steven H. Dynamical Systems We study a system of $N$ interacting particles moving on the unit sphere in $d$-dimensional space. The particles are self-propelled and coupled all to all, and their motion is heavily overdamped. For $d=2$, the system reduces to the classic Kuramoto model of coupled oscillators; for $d=3$, it has been proposed to describe the orientation dynamics of swarms of drones or other entities moving about in three-dimensional space. Here we use group theory to explain the recent discovery that the model shows low-dimensional dynamics for all $N \ge 3$, and to clarify why it admits the analog of the Ott-Antonsen ansatz in the continuum limit $N \rightarrow \infty$. The underlying reason is that the system is intimately connected to the natural hyperbolic geometry on the unit ball $B^d$. In this geometry, the isometries form a Lie group consisting of higher-dimensional generalizations of the Möbius transformations used in complex analysis. Once these connections are realized, the reduced dynamics and the generalized Ott-Antonsen ansatz follow immediately. This framework also reveals the seamless connection between the finite and infinite-$N$ cases. Finally, we show that special forms of coupling yield gradient dynamics with respect to the hyperbolic metric, and use that fact to obtain global stability results about convergence to the synchronized state. |
| title | The Kuramoto model on a sphere: Explaining its low-dimensional dynamics with group theory and hyperbolic geometry |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/1907.07150 |