Geometry and conformal invariants of Kuramoto Oscillator Networks, relating finite N and continuum descriptions

Kuramoto oscillators governed by the same ODE advance in time under the same 3-parameter Möbius transformation. So the dynamics of a state of N identical Kuramoto oscillators is constrained to its G–orbit under the Möbius group action, which is a 3D manifold in T^N. The N – 3 independent cross-ratio's of the oscillator phases (e^iθ_j , j = 1, ..., N) are invariants of conformal Möbius transformations. For phase models one can project to 2D orbits and then the oscillator dynamics map to flows on the Poincaré disk. The geometric implications of this correspondence was studied in Ref 1. Here we consider the continuum limit. Previous work identified Poisson densities as the continuum limit of states on the finite N splay orbit. We extend this identification to more complicated densities for which the dynamics preserve new conformal invariants that have a geometric interpretation. Finally we consider continuum networks with distributions of natural frequencies and identify models for which the asymptotic collective order-parameter dynamics correspond to those on the Ott-Antonsen manifold as well as models for which they do not.

[1] Hyperbolic geometry of Kuramoto oscillator networks, B Chen, JR Engelbrecht, R Mirollo, J Phys A: Math & Theor, 50 (35), 355101.