Periodic perturbations of oscillators on the plane

Abstract

A review of the results of research carried out in the current century at the Department of Differential Equations of St. Petersburg State University is presented. We study the problem of stability of the zero solution of a second-order equation describing periodic perturbations of an oscillator with a nonlinear restoring force under reversible and conservative perturbations. Such perturbations are related to transcendental perturbations, in which, in order to solve the problem of stability, it is necessary to take into account all the terms of the expansion of the right side of the equation into a series. The problem of stability under transcendental perturbations was posed in 1893 by A. M. Lyapunov. The results presented in this article on the stability of the oscillator were carried out using the KAM-theory methods: perturbations of the oscillator with infinitely small and infinitely large oscillation frequencies are considered; conditions for the presence of quasi-periodic solutions in any neighborhood of the time axis are given, from which follows the stability (not asymptotic) of the zero solution of the perturbed equation; stability conditions are given for the zero solution of a Hamiltonian system with two degrees of freedom, the unperturbed part of which is described by a pair of oscillators (in this case, conservative perturbations are considered).