Evolution of slow variables in a priori unstable Hamiltonian systems

We study diffusion phenomena in a priori unstable (initially hyperbolic) Hamiltonian systems. These systems are perturbations of integrable ones, which have a family of hyperbolic tori. We prove that in the case of two and a half degrees of freedom the action variable generically drifts (i.e. changes on a trajectory by a quantity of order one). Moreover, there exists a trajectory such that the velocity of this drift is ε/logε, where ε is the parameter of the perturbation.