In this paper, we consider two singular planar differential systems
which can describe the evolution of the optical pulse width and chirp
for the so-called dispersion-managed solitons. Based on the method of
third order approximation in combination with some quantitative
information obtained by the upper-lower solutions method and the
averaging method, some results on the existence and Lyapunov stability
of the periodic solutions are obtained. Moreover, the formula of the
first twist coefficient and a stability criterion of a nonlinear
differential equation are also established.