  The Kuramoto--Sakaguchi model is a modification of the well-known Kuramoto
model that adds a phase-lag paramater, or "frustration" to a network of
phase-coupled oscillators. The Kuramoto model is a flow of gradient type, but
adding a phase-lag breaks the gradient structure, significantly complicating
the analysis of the model. We present several results determining the stability
of phase-locked configurations: the first of these gives a sufficient condition
for stability, and the second a sufficient condition for instability. (In fact,
the instability criterion gives a count, modulo 2, of the dimension of the
unstable manifold to a fixed point and having an odd count is a sufficient
condition for instability of the fixed point.) We also present numerical
results for both small and large collections of Kuramoto--Sakaguchi
oscillators.
