Improved Estimates on the Existence of Invariant Tori for Hamiltonian Systems

The existence of invariant tori in nearly--integrable Hamiltonian systems is investigated. We focus our attention on a particular one--dimensional, time--dependent model, known as the \sl forced pendulum. \rm We present a KAM algorithm which allows to derive explicit estimates on the perturbing parameter ensuring the existence of invariant tori. Despite previous proofs of KAM theorem, we introduce some technical novelties which allow to provide results in good agreement with the experimental break--down threshold. In particular, we have been able to prove the existence of the golden torus with frequency ${{\sqrt{5}-1}\over 2}$ for values of the perturbing parameter equal to $92.4\%$ of the numerical threshold.