Periodic Perturbations of Oscillators on a Plane

Abstract

The results of research carried out in the 21st century at the Department of Differential Equations of St. Petersburg State University are reviewed. The subject of study is the problem of stability of the zero solution to the second-order equation describing periodic perturbations of the oscillator with a nonlinear restoring force under reversible and conservative perturbations. Such perturbations are classified as transcendental perturbations, for which the solution of the problem of stability requires taking into account all terms in the expansion in a series of the right-hand side of the equation. The problem of stability under transcendental perturbations was formulated in 1893 by A.M. Lyapunov. The results regarding stability presented in this review were obtained using the methods of KAM theory: perturbations of the oscillator with infinitely small and infinitely large oscillation frequencies were considered; conditions for the existence of quasi-periodic solutions in any vicinity of the time axis are determined, from which the stability (not asymptotic) of the zero solution of the perturbed equation follows; and the conditions are found for the stability of the zero solution for 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).