Normal form construction for nearly-integrable systems with dissipation

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