APS -APS March Meeting 2017 - Event - Geometry in a dynamical system without space: Hyperbolic Geometry in Kuramoto Oscillator Systems

摘要

Kuramoto oscillator networks have the special property that their time evolution is constrained to lie on 3D orbits of the M"obius group acting onthe $N$-fold torus $T^N$ which explains the $N-3$ constants of motiondiscovered by Watanabe and Strogatz. The dynamics for phase models can befurther reduced to 2D invariant sets in $T^{N-1}$ which have a natural geometryequivalent to the unit disk $Delta$ with hyperbolic metric. We show that theclassic Kuramoto model with order parameter $Z_1$ (the first moment of theoscillator configuration) is a gradient flow in this metric with a unique fixedpoint on each generic 2D invariant set, corresponding to the hyperbolicbarycenter of an oscillator configuration. This gradient property makes thedynamics especially easy to analyze. We exhibit several new families ofKuramoto oscillator models which reduce to gradient flows in this metric; someof these have a richer fixed point structure including non-hyperbolic fixedpoints associated with fixed point bifurcations.