  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 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,
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 in sync, as well as multiple families of neutrally stable
asynchronous states or closed orbits, in which no two oscillators are in sync.
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.
