Conformal Group Actions on Generalized Kuramoto Oscillators

This paper unifies the recent results on generalized Kuramoto Model reductions. Lohe took a coupled system of $N$ bodies on $S^d$ governed by the Kuramoto equations $\dot{x_i}=\mathrm{\Omega }{x}_i+X-⟨x_i,X⟩x_i$ and used the method of Watanabe and Strogatz to reduce this system to $d + \frac{d(d-1)}{2}$ equations. Using a model of rigid rotations on a sphere as a guide, we show that the reduction is described by a smooth path in the Lie group of conformal transformations on the sphere, which is diffeomorphic to $SO(d)\times D^d$. Seeing the reduction this way allows us to apply geometric and topological reasoning in order to understand qualitative behavior of the Kuramoto Model.