In this paper, we investigate perturbations of linear integrable Hamiltonian
systems, with the aim of establishing results in the spirit of the KAM theorem
(preservation of invariant tori), the Nekhoroshev theorem (stability of the
action variables for a finite but long interval of time) and Arnold diffusion
(instability of the action variables). Whether the frequency of the integrable
system is resonant or not, it is known that the KAM theorem does not hold true
for all perturbations; when the frequency is resonant, it is the Nekhoroshev
theorem which does not hold true for all perturbations. Our first result deals
with the resonant case: we prove a result of instability for a generic
perturbation, which implies that the KAM and the Nekhoroshev theorem do not
hold true even for a generic perturbation. The case where the frequency is
non-resonant is more subtle. Our second result shows that for a generic
perturbation, the KAM theorem holds true. Concerning the Nekhrosohev theorem,
it is known that one has stability over an exponentially long interval of time,
and that this cannot be improved for all perturbations. Our third result shows
that for a generic perturbation, one has stability for a doubly exponentially
long interval of time. The only question left unanswered is whether one has
instability for a generic perturbation (necessarily after this very long
interval of time)
