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The Kuramoto model on a sphere: Explaining its low-dimensional dynamics with group theory and hyperbolic geometry
Max Lipton, Renato Mirollo, Steven H. Strogatz
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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 ≥3, and to clarify why it admits the analog of the Ott-Antonsen ansatz in the continuum limit N →∞. The underlying reason is that the system is intimately connected to the natural hyperbolic geometry on the unit ball Bᵈ. In this geometry, the isometries form a Lie group consisting of higher-dimensional generalizations of the M\"obius 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.
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