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1. Möbius Transformation and Hyperbolic Geometry in Kuramoto Oscillators
In the over-damped regime of an array of superconducting Josephson junctions, the effective degrees of freedom are captured by so-called Kuramoto model where angular variables are interacting via trigonometric functions of relative phase differences.
The system of N identical Kuramoto oscillators can be formulated as complex Riccati equations. And the dynamics are invariant under 3D Möbius transformations on the unit disk, which are elements of the projective linear group PGL(2,R). The evolution of N phase oscillators is equivalent to a 3D flow in the group space.
2. Group Orbits and Order Parameters in Extended Kuramoto Systems
Kuramoto oscillator networks have the special property that their time evolution is constrained to lie on 3D orbits of the Möbius group acting on the N-fold torus T^{N} which explains the N-3 constants of motion discovered by Watanabe and Strogatz. The dynamics for phase models can be further reduced to 2D invariant sets in T^{N-1} which have a natural geometry equivalent to the unit disc with hyperbolic metric.
We show that the classic Kuramoto model with order parameter Z_{1} (the first moment of the oscillator configuration) is a gradient flow in this metric with a unique fixed point on each generic 2D invariant set, corresponding to the hyperbolic barycenter of an oscillator configuration. This gradient property makes the dynamics especially easy to analyze. We exhibit several new families of Kuramoto oscillator models which reduce to gradient flows in this metric; some of these have a richer fixed point structure including non-hyperbolic fixed points associated with fixed point bifurcations.
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