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

Abstract

The paper considers small periodic regular and singular perturbations of a system, whose conservative part is an oscillator with cubic restoring force. The smallness of perturbations is due to both the smallness of the neighborhood of equilibrium and the presence of a small parameter. In the absence of a small parameter, we obtain conditions for Lyapunov stability of the equilibrium position. If a small parameter is present, we derive (both for regular and singular perturbations) an equation whose positive roots are in correspondence with invariant two-dimensional tori of the perturbed system.