  The Chapman-Enskog method of kinetic theory is applied to two problems of
synchronization of globally coupled phase oscillators. First, a modified
Kuramoto model is obtained in the limit of small inertia from a more general
model which includes ``inertial'' effects. Second, a modified Chapman-Enskog
method is used to derive the amplitude equation for an O(2) Takens-Bogdanov
bifurcation corresponding to the tricritical point of the Kuramoto model with a
bimodal distribution of oscillator natural frequencies. This latter calculation
shows that the Chapman-Enskog method is a convenient alternative to normal form
calculations.
