Separatrices Splitting for the Rapidly Forced Pendulum

The equation of the rapidly forced pendulum $$\ddot x + \sin (x) = \delta {\varepsilon ^p}\sin (t/\varepsilon ),$$ p > 5, is considered. It is proven that Poincare-Arnold-Melnikov method gives the right asymptotic formula for the separatrices splitting angle for the case of a small e.