On Periodic Motions of a Nearly Autonomous Hamiltonian System in the Occurrence of Double Parametric Resonance

Abstract

Motions of a 2π-periodic nearly autonomous Hamiltonian system with two degrees of freedom in the neighborhood of the equilibrium position are examined. It is assumed that the Hamiltonian of the system depends on three parameters, namely, ε, α, and β, and the system is an autonomous system if ε = 0. Let a double parametric resonance, i.e., a situation when one of the frequencies of small linear oscillations of the system in the neighborhood of the equilibrium position is an integer number and the other one is a half-integer number, takes place in an unperturbed (ε = 0) system for some α and β values. For sufficiently small, but nonzero, ε values in a small neighborhood of the resonance point considered with a fixed resonant value of one parameter (β), the issue of the existence, bifurcations, and stability in the linear approximation of periodic motions of the system is resolved. In the occurrence of multiple resonances of the type under study, periodic motions of a dynamically symmetrical satellite in the neighborhood of its stationary rotation (cylindrical precession) in a slighty elliptical orbit are constructed and linear and nonlinear analyses of their stability are carried out.