Scaling law for the critical function of an approximate renormalization

Abstract

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.