The Kuramoto model on a sphere: Explaining its low-dimensional dynamics with group theory and hyperbolic geometry

We study a system of N identical 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 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.