Parabolic fixed points and stability criteria for nonlinear Hill's equation

We discuss the stability of parabolic fixed points of area-preserving mappi ngs and obtain a new proof of a criterion due to Simo. These results are em ployed to discuss the stability of the equilibrium of certain periodic diff erential equations of newtonian type. An example is the pendulum of variabl e length. In this class of equations the First Lyapunov's Method does not a pply but in many cases the stability can be characterized in terms of the v ariational equation.