The paper is centered on the nonlinear differential equation \[ \ddot x+ x^3= X(x,\dot x,t,\varepsilon) \] with a small parameter \(\varepsilon>0\). The function \(X\) is assumed to be sufficiently smooth for \(t\in\mathbb{R}\), \(|x|<x^*\), \(|\dot x|< x^*\), and \(\varepsilon\in[0, \varepsilon^*)\), quasiperiodic in \(t\) with basic frequencies \(\omega_1,\dots,\omega_m\), and infinitesimal of fourth-order with respect to the variables \(x\), \(\dot x\), and \(\varepsilon\).
In the general case of the equation, the author establishes the existence of its invariant surface such that motions on it are many-frequency oscillations with frequencies \(\omega_1,\dots,\omega_m\) and the additional one is infinitesimal of the order of \(\sqrt\varepsilon\). This invariant surface is proved to be asymptotically stable that means the convergence of any solution from a small neighborhood to a solution disposed on it. In conclusion, analogous results are obtained for conservative and invertible equations being special cases of the original one.