Lower and upper bounds for the splitting of separatrices of the pendulum under a fast quasiperiodic forcing

Abstract

Quasiperiodic perturbations with two frequencies $(1/\varepsilon ,\gamma /\varepsilon )$ of a pendulum are considered, where $\gamma $ is the golden mean number. We study the splitting of the three-dimensional invariant manifolds associated to a two-dimensional invariant torus in a neighbourhood of the saddle point of the pendulum. Provided that some of the Fourier coefficients of the perturbation (the ones associated to Fibonacci numbers) are separated from zero, it is proved that the invariant manifolds split for $\varepsilon $ small enough. The value of the splitting, that turns out to be ${\rm O} (\exp (-{\rm const} /\sqrt{\varepsilon }) )$, is correctly predicted by the Melnikov function.