Regular and singular periodic perturbations of an oscillator with cubic restoring force (Q1759549)

The paper is devoted to the study of the system \[ \begin{aligned} \dot x &= y + X_0(x,y,z,t) + \varepsilon X(x,y,z,t, \varepsilon), \\ \dot y &= -x^3 + Y_0(x,y,z,t) + \varepsilon Y(x,y,z,t,\varepsilon), \\ \dot z &= Az + Z_0(x,y,z,t) + \varepsilon Z(x,y,z,t,\varepsilon), \end{aligned}\tag{1} \] where \(x, y \in \mathbb{R}, z \in \mathbb{R}^n, \varepsilon > 0,\) all eigenvalues of the constant matrix \(A\) have nonzero real parts, the functions \(X_0, Y_0, Z_0\) and \(X,Y,Z\) are infinitely often differentiable with respect to \(x,y,z,\varepsilon\) and \(T\)-periodic in \(t\) in some neighborhood of \(x=0\), \(y =0\), \(z=0\), \(\varepsilon=0\). If \(\varepsilon = 0\), sufficient conditions for the asymptotic stability of the zero-solution of (1) are given. For small \(\varepsilon > 0\), it is proved (under some addition assumptions) that the system has a two-dimensional invariant torus. The authors also obtain conditions for the existence of a two-dimensional torus for a singularly perturbed system in the variables \(x,y,z,\xi\), where the equations for \(x,y,z\) have the form of system (1), and, for \(\xi \in \mathbb{R}^m\), the system has the form \[ \varepsilon \dot\xi = B\xi + \Xi_0(x,y,z,\xi,t) + \varepsilon \Xi_0(x,y,z,\xi,t,\varepsilon), \] where the constant \((m\times m)\)-matrix \(B\) has eigenvalues with nonzero real parts.