Improved estimates on the existence of invariant tori for Hamiltonian systems

by Alessandra Celletti, Antonio Giorgilli and Ugo Locatelli

**Abstract:**
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}.
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 KAM theorem
which allow us 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 $92\,\%$ of the numerical threshold,
thus significantly improving the previous calculations.

preprint (1999)

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