We consider a class of Hamiltonians with three degrees of freedom that can be
mapped into quasi-periodically driven pendulums. The purpose of this paper is
to determine the threshold of the break-up of invariant tori with a specific
frequency vector. We apply two techniques: the frequency map analysis and
renormalization-group methods. The renormalization transformation acting on a
Hamiltonian is a canonical change of coordinates which is a combination of a
partial elimination of the irrelevant modes of the Hamiltonian and a rescaling
of phase space around the considered torus. We give numerical evidence that the
critical coupling at which the renormalization transformation starts to diverge
is the same as the value given by the frequency map analysis for the break-up
of invariant tori. Furthermore, we obtain by these methods numerical values of
the threshold of the break-up of the last invariant torus.Comment: 18 pages, 4 figure
