For systems as in (1), the unperturbed Hamiltonian possesses smooth families of invariant tori that have hyperbolic stable and unstable manifolds; such systems are referred to as a priori unstable. For system as in (2), the phase space of the unperturbed Hamiltonian is foliated by Lagrangian invariant tori; such systems are referred to as a priori stable. In either case, the Arnold diffusion problem asserts that, for `generic' systems, there exists ρ ρ > 0 such that for every ϵ ϵ > 0 sufficiently small there exist solution curves for which