On Nonlinear Oscillations of a Near-Autonomous Hamiltonian System in One Case of Integer Nonequal Frequencies

This paper is concerned with the motions of a near-autonomous two-degree-of-freedom Hamiltonian system, $2\pi$-periodic in time, in a neighborhood of a trivial equilibrium. It is assumed that in the autonomous case, in the region where only necessary (which are not sufficient) conditions for the stability of this equilibrium are satisfied, for some parameter values of the system one of the frequencies of small linear oscillations is equal to two and the other is equal to one. An analysis is made of nonlinear oscillations of the system in a neighborhood of this equilibrium for the parameter values near a resonant point of parameter space. The boundaries of the parametric resonance regions are constructed which arise in the presence of secondary resonances in the transformed linear system (the cases of zero frequency and equal frequencies). The general case and both cases of secondary resonances are considered; in particular, the case of two zero frequencies is singled out. An analysis is made of resonant periodic motions of the system that are analytic in integer or fractional powers of the small parameter, and conditions for their linear stability are obtained. Using KAM theory, two- and three-frequency conditionally periodic motions (with frequencies of different orders in a small parameter) are described.