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 forced pendulum . We present a KAM algorithm which allows us to derive explicit estimates on the perturbing parameter ensuring the existence of invariant tori. Moreover, we introduce some technical novelties in the proof of the KAM theorem which allow us to provide results in good agreement with the experimental breakdown threshold. In particular, we have been able to prove the existence of the golden torus with frequency ½((5)1/2 -1) for values of the perturbing parameter equal to 92% of the numerical threshold, thus significantly improving the previous calculations.