  We construct an approximate renormalization for Hamiltonian systems with two
degrees of freedom in order to study the break-up of invariant tori with
arbitrary frequency. We derive the equation of the critical surface of the
renormalization map, and we compute the scaling behavior of the critical
function of one-parameter families of Hamiltonians, near rational frequencies.
For the forced pendulum model, we find the same scaling law found for the
standard map in [Carletti and Laskar, preprint (2000)]. We discuss a conjecture
on the link between the critical function of various types of forced pendulum
models, with the Bruno function.
