  We consider a dissipative vector field which is represented by a
nearly-integrable Hamiltonian flow to which a non symplectic force is added, so
that the phase space volume is not preserved. The vector field depends upon two
parameters, namely the perturbing and dissipative parameters, and by a drift
function. We study the general case of an l-dimensional, time-dependent vector
field. Assuming to start with non-resonant initial conditions, we prove the
stability of the variables which are actions of the conservative system
(namely, when the dissipative parameter is set to zero) for exponentially long
times. In order to construct the normal form, a suitable choice of the drift
function must be performed. We also provide some simple examples in which we
construct explicitly the normal form, we make a comparison with a numerical
integration and we compute theoretical bounds on the parameters as well as we
give explicit stability estimates.
