  We provide an analysis of the classic Kuramoto model of coupled nonlinear
oscillators that goes beyond the existing results for all-to-all networks of
identical oscillators. Our work is applicable to oscillator networks of
arbitrary interconnection topology with uncertain natural frequencies. Using
tools from spectral graph theory and control theory, we prove that for
couplings above a critical value, the synchronized state is locally
asymptotically stable, resulting in convergence of all phase differences to a
constant value, both in the case of identical natural frequencies as well as
uncertain ones. We further explain the behavior of the system as the number of
oscillators grows to infinity.
