On stability of the vector Liénard equation with nonstationary perturbations (Q1975803)

The author exposes a method for studying the stability of solutions to nonautonomous systems in critical cases. The influence is studied of nonstationary perturbations on the essentially nonlinear system of differential equations \[ \ddot X + \frac{\partial F}{\partial X}\dot X + \frac{\partial G}{\partial X} = 0, \] where \(X\) is the \(n\)-dimensional vector of unknown functions, and the scalar function \(G\) and the components of the \(n\)-dimensional vector \(F(x)\) are continuously differentiable homogeneous functions of order \(\mu = 1\) and \(\nu =1\) respectively, where \(\mu\) and \(\nu\) are rationals with odd denominators such that \(\mu >1\) and \(\nu>1\). The author defines classes of perturbations which preserve the asymptotic stability and instability of the zero solution. It is proven that the order of perturbations may be less than that of the functions on the right-hand sides of the original equations.