  We study the stability of a vector field associated to a nearly-integrable
Hamiltonian dynamical system to which a dissipation is added. Such a system is
governed by two parameters, named the perturbing and dissipative parameters,
and it depends on a drift function. Assuming that the frequency of motion
satisfies some resonance assumption, we investigate the stability of the
dynamics, and precisely the variation of the action variables associated to the
conservative model. According to the structure of the vector field, one can
find linear and exponential stability times, which are established under
smallness con- ditions on the parameters. We also provide some applications to
concrete examples, which exhibit a linear or exponential stability behavior.
