AN ASYMPTOTIC EXPRESSION FOR THE SPLITTING OF SEPARATRICES OF THE RAPIDLY FORCED PENDULUM

The measure of the splitting of the separatrices of the rapidly forced pendulum x double over dot + sin x = mu sin t/epsilon, is considered as a model problem that has been studied by different authors. Here epsilon, mu are small parameters, epsilon > 0, but otherwise independent. The following formula for the angle alpha between separatrices is established [GRAPHICS] This formula is also valid for the particular case mu = epsilon(p), with p > 0, epsilon > 0, and agrees with the one provided by the first order Poincare-Melnikov theory that cannot be applied directly, due to the exponentially small dependence of alpha on the parameter epsilon.