Functions that determine stability of rational rotations of a near symmetric satellite

The satellite oscillation equation is a nonlinear second order ordinary differential equation with two parameters, one of which is supposed to be small and the second (the orbit eccentricity e) varies up to a singular value. We discuss the computation of the leading coefficient of the averaged equation (i.e. first approximation of the normal form) in cases of integer and rational mean angular velocity. A regularization near the singular value e=1 is described. An effective qualitative control of the computations is provided by comparing numeric results with control asymptotics obtained by the saddle point method.