On Nonlinear Oscillations of a Time-Periodic Hamiltonian System at a 2:1:1 Resonance

Abstract

We consider the motions of a near-autonomous Hamiltonian system $2\pi$-periodic in time, with two degrees of freedom, in a neighborhood of a trivial equilibrium. A multiple parametric resonance is assumed to occur for a certain set of system parameters in the autonomous case, for which the frequencies of small linear oscillations are equal to two and one, and the resonant point of the parameter space belongs to the region of sufficient stability conditions. Under certain restrictions on the structure of the Hamiltonian of perturbed motion, nonlinear oscillations of the system in the vicinity of the equilibrium are studied for parameter values from a small neighborhood of the resonant point. Analytical boundaries of parametric resonance regions are obtained, which arise in the presence of secondary resonances in the transformed linear system (the cases of zero frequency and equal frequencies). The general case, for which the parameter values do not belong to the parametric resonance regions and their small neighborhoods, and both cases of secondary resonances are considered. The question of the existence of resonant periodic motions of the system is solved, and their linear stability is studied. Two- and three-frequency conditionally periodic motions are described. As an application, nonlinear resonant oscillations of a dynamically symmetric satellite (rigid body) relative to the center of mass in the vicinity of its cylindrical precession in a weakly elliptical orbit are investigated.