Analysis of the Strong and Weak Monotonic External Stability of the Resonance in a Perturbed Dynamical System

Abstract

A perturbed dynamical system involving two ordinary differential equations is under review. Whereupon, the differential equation for determining the fast phase contains the ratio of the two frequencies. When these frequencies coincide for a long time, a resonance is implemented in this system. The aim of this paper is to obtain the conditions of monotonic external stability and instability of this resonance. The sufficient conditions for the external stability and instability of the resonance defined in this paper assume that the signs of the analyzed derivatives remain unchanged in the non-resonant section of the change in the independent variable. This paper gives a new classification of the phenomenon of external stability of resonance, which includes weak, linear, and strong stability. It should be noted that the conditions of monotonic external stability and instability of the resonance presented in this paper can be used in various scientific and technological problems, in which resonances are observed. Particularly, the concluding part of the paper considers the application of the results obtained within the framework of the problem of the perturbed motion of a rigid body relative to a fixed point.