  We analyze the breakup of invariant tori in Hamiltonian systems with two
degrees of freedom using a combination of KAM theory and renormalization-group
techniques. We consider a class of Hamiltonians quadratic in the action
variables that is invariant under the chosen KAM transformations, following the
approach of Thirring. The numerical implementation of the transformation shows
that the KAM iteration converges up to the critical coupling at which the torus
breaks up. By combining this iteration with a renormalization, consisting of a
shift of resonances and rescalings of momentum and energy, we obtain a much
more efficient method that allows to determine the critical coupling with high
accuracy. This transformation is based on the physical mechanism of the breakup
of invariant tori. We show that the critical surface of the transformation is
the stable manifold of codimension one of a nontrivial fixed point, and we
discuss its universality properties.
