  We show the existence of drifting orbits for certain perturbations of
non-convex Hamiltonian systems with several degrees of freedom. These orbits
remain in the vicinity of resonant surfaces where the action variables can
undergo changes $O(1)$ infinitely often although the size of perturbations
$O(\epsilon)$ can be arbitrarily small. The first drifts occur in a period of
time $O(1/\epsilon)$ and then reoccur with frequencies independent of
$\epsilon$. We also perform numerical simulations to compare the effects of two
conditions for instability in two four-dimensional examples with random
parameters.
