Dynamics of the Kuramoto-Sakaguchi oscillator network with asymmetric order parameter.

We study the dynamics of a generalized version of the famous Kuramoto-Sakaguchi coupled oscillator model. In the classic version of this system, all oscillators are governed by the same ordinary differential equation (ODE), which depends on the order parameter of the oscillator configuration. The order parameter is the arithmetic mean of the configuration of complex oscillator phases, multiplied by some constant complex coupling factor. In the generalized model, we consider that all oscillators are still governed by the same ODE, but the order parameter is allowed to be any complex linear combination of the complex oscillator phases, so the oscillators are no longer necessarily weighted identically in the order parameter. This asymmetric version of the K-S model exhibits a much richer variety of steady-state dynamical behavior than the classic symmetric version; in addition to stable synchronized states, the system may possess multiple stable (N-1,1) states, in which all but one of the oscillators are synchronized, as well as multiple families of neutrally stable states or closed orbits, in which no two oscillators are synchronized. We present an exhaustive description of the possible steady state dynamical behaviors; our classification depends on the complex coefficients that determine the order parameter. We use techniques from group theory and hyperbolic geometry to reduce the dynamic analysis to a 2D flow on the unit disc, which has geometric significance relative to the hyperbolic metric. The geometric-analytic techniques we develop can in turn be applied to study even more general versions of Kuramoto oscillator networks.