On the Motions of a Nearly Autonomous Hamiltonian System at Parameter Values Close to the Boundary of Stability Regions of the Limiting Autonomous Problem

The paper studies the motions of a nearly autonomous, Hamiltonian system $2\pi$-periodic in time, with two degrees of freedom, in the neighborhood of a trivial equilibrium. The values of the parameters are considered near the boundary of the stability region of this equilibrium that corresponds to the zero frequency case in the limiting autonomous problem. The cases are distinguished when the other frequency is nonresonant (not equal to an integer or halfinteger number) and resonant, i. e., a multiple parametric resonance is realized in the system. The stability and instability regions (parametric resonance regions) of the trivial equilibrium of the system are obtained. The question of the existence in its neighborhood of analytic (in integer or fractional powers of a small parameter) resonant periodic motions, their number and linear stability is resolved. Earlier, similar results for the studied cases of multiple parametric resonances have been obtained in the sections of the parameter space corresponding to a fixed (resonant) value of one of the parameters. In this paper, complete neighborhoods of resonance points in the parameter space are considered. As expected, this leads to complication of both the stability diagram of the trivial equilibrium and the distribution and character of the stability of the periodic solutions under study. For each resonance case, an algorithm for analyzing these motions is developed. Differences and similarities in the properties of the system for different resonant cases are established, parallels are found with multiple resonance cases of another type, arising in the systems with another structure of the perturbing part of the Hamiltonian function.