On nonlinear oscillations of time-periodic Hamiltonian systems in the presence of double fourth-order resonances

The motion of a time-periodic Hamiltonian system with two degrees of freedom in a neighborhood of a linearly stable equilibrium is investigated. A double fourth-order resonance (of fundamental and combination types) is assumed to be realized in the system, or the parameters of the problem are close to the resonance values. The problem of the existence and number of resonant periodic motions in a small neighborhood of the equilibrium is investigated; the conditions for their stability in the linear approximation are analyzed. The results are applied to the problem of the motion of a dynamically symmetric satellite (a rigid body) near its stationary rotation (cylindrical precession) in the central Newtonian gravitational field in an elliptical orbit of arbitrary eccentricity. (paper)