THE PROBLEMS related to nonlinear forced oscillations that engineers are faced with, can generally be reduced to the study of smooth differential systems such as i = X(x, 2, t) depending on a parameter 2, 2x-periodic with respect to time. An extensive literature, theoretical as well as numerical and experimental, deals with the steady state responses of these systems, specially with time periodic solutions. The non uniqueness of these solutions, their change of nature and stability with parameter values are now well known phenomena. The existence, for some values of i, of bounded but not periodic steady state solutions is still being investigated. Such oscillations have experimentally been observed for mechanical systems [I] and for electrical systems [2] and on analog computer models [3]. These oscillations can reach large amplitudes and lead to defective behaviour of the physical system [ 11. So it is of interest to know the domain of initial values and parameter values for which they appear. Let us point out that we are dealing with cases where the system obtained by suppressing the external periodic excitation has no periodic solutions. The work of Moser [4,5] and Cartwright [6] deal with the general theory of these oscillations. A formal approach by the Ritz-Galerkin method can be found in [3,7]. Using a result obtained by Sacker [S] and, in an independent way, by Ruelle and Takens [9] the existence of a torus of solutions is shown for the differential system provided that a specific bifurcating situation happens (see assumptions H,, H,, H, below). The observed oscillations correspond to a solution lying on the torus. Lanford [lo] presents Ruelle and Takens’ results in a form which is, according to us, suitable to study the nonlinear oscillations. In a recent paper [l I] Iooss applies and develops these results for a system of equations of Navier-Stokes type. Then he proves that solutions which bifurcate from the basic time-periodic solution, for some I = i, lie on an invariant torus (the solutions with initials values on the torus lie on the torus) which may be attracting or repelling (for solutions with initial values near the torus). The basic flow looses its stability for 1 = ilc. Following the last author it is possible to carry out numerically the theoretical results as it is shown in Section 4 by an example given by the physic of vibrations. It is not possible to specify the nature of the solutions lying on the torus (quasi periodic? almost