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

Stability estimates of nearly-integrable systems with dissipation and non-resonant frequency