Stable odd solutions of some periodic equations modeling satellite motion

A new stability criterion is proved for second-order differential equations with symmetries in terms of the coefficients of the expansion of the nonlinearity up to the third order. Such a criterion provides solutions of twist type, which are Lyapunov-stable solutions with interesting dynamical properties. This result is connected with the existence of upper and lower solutions of a Dirichlet problem and applied to a known equation which model the planar oscillations of a satellite in an elliptic orbit, giving an explicit region of parameters for which there exists a Lyapunov-stable solution.