0,\tau,N$ such that $J_0\in\Gamma(\gamma,\tau,N)$, then there exists a constant $\epsilon_*$, such that, if $N>K_*\left|\ln\eps_*\right|=K(\epsilon_*)$, and $\exp(-{N}/{K_*})<\eps<\eps_*$, then (1) has a stable invariant torus close to the torus $\toro_0$. } {\it Remark 3.}\ The condition $J_0\in\Gamma(\gamma,\tau,N)$ is surely fulfilled if $\omega (J_0)$ is diophantine; but it can be fulfilled also if $\omega (J_0)$ satisfies some very high order resonance relations. To clarify this point assume that there exists $K_{\sharp}$, such that $\omega(J_0)$ satisfies some resonance relation of order $K_\sharp$ (namely that $\omega(J_0)\cdot k=0$ for some $k\in Z^m$ with $|k|=K_{\sharp}$), but there exist $\tau,\gamma$ such that $J_0\in\Gamma(\gamma,\tau,K_{\sharp}-1)$, then our theorem gives no results for very small $\eps$, namely for $0<\eps<\exp(-K_{\sharp}/K_*)$, but ensures the existence of an invariant torus for $\eps$ satisfying $\exp(-(K_{\sharp}-1)/K_*)<\eps<\eps_*$. \vskip 10pt {\it Remark 4.}\ The proof of our theorem does not require any nondegeneracy condition for the dependence of $\omega$ on the actions $I$; moreover, the result we obtain are quite different from those that one can expect to obtain by KAM techniques. Indeed, KAM techniques in hamiltonian systems allow to ensure that there exists a large Cantor-like set ${\cal S}$ in the action space such that, corresponding to any $I \in {\cal S}$ there exists an invariant torus of the system; the set ${\cal S}$ usually depends on $\eps$. In the present nonconservative case, one can expect KAM techniques would allow to prove (under nondegeneracy conditions) that, if the solution of an equation of the kind $\Fm_{\eps} (J )=0$ is in a suitable Cantor-like set, then correspondingly there exists an invariant torus for (1). If this Cantor set actually depends on $\eps$, then typically an invariant torus exists for a Cantor set of values of $\eps$. On the contrary we obtain existence of the torus for $\eps$ in an open set. {\it Remark 5.}\ Finally, we remark that the condition that $\t0$ is attractive (obviously, the repulsive case would be equivalent) comes into play through the negativity of nonzero Lyapounov exponents; should we have a generic transversally hyperbolic $\t0$, we should consider negative and positive Lyapounov exponents, and correspondingly a stable and an unstable manifold; we should then consider their perturbations, and the intersection of the perturbed manifolds, in order to obtain the perturbed invariant torus. Thus, we expect that the present scheme could be extended to the general case along these lines; however, up to now we have not performed the computations. \bigskip \bigskip\vfill {\bf Acknowledgements} {First of all, we would like to warmly thank the referee for pointing out an error in the first version of the paper; this led to the detailed proof given in ref. [7]. This work was performed during exchange visits; in particular, we acknowledge the support of a LU-PAS Research Committee grant, which made possible the visit of D.B. in Loughborough. The work was completed while G.G. was visiting the I.H.E.S.: thanks to its Director and all the personnel for the warm hospitality; special thanks to Cecile Gourgues for help with the French version. The work has been also partially supported by the grant EC contract ERBCHRXCT940460 for the project ``Stability and universality in classical mechanics".} \vfill\eject {\bf References} [1] V.I.~Arnold, V.V.~Kozlov and A.I.~Neishtadt: {\it Mathematical aspects of classical and celestial mechanics}; In V.I.~Arnold (ed.): {\it Dynamical Systems III}, Encyclopaedia of mathematical sciences, vol.~3, Springer, Berlin 1988 [2] N.N.~Bogoljubov, Ju.A.~Mitropoliskii and A.M.~Samolienko: {\it Methods of Accelerated Convergence in Nonlinear Mechanics}; Springer, Berlin 1976 [3] K.~Yagasaki: ``{Chaotic motions near homoclinic manifolds and resonant tori in quasi-periodic perturbations of planar Hamiltonian-systems}''; {\it Physica D} {\bf 69}, 232-269 (1993) [4] N.~Fenichel: ``Persistence and smoothness of invariant manifolds for flows''; {\it Ind. Univ. Math. J} {\bf 21}, 193--226 (1971) [5] M.W.~Hirsch, C.C.~Pugh and M.~Shub: {\it Invariant Manifolds}; Lecture Notes Mathematics {\bf 583}, Springer, Berlin 1977 [6] M.~Andreolli, D.~Bambusi and A.~Giorgilli: ``{On a weakened form of the averaging principle in multifrequency systems}''; {\it Nonlinearity} {\bf 8}, 283--293 (1995). [7] D.~Bambusi and G.~Gaeta: ``Proof of persistence of invariant tori in nonhamiltonian perturbations of integrable systems''; Preprint {\tt mp\_arc 97-***} (1997) \bye