On Periodic Motions of a Nonautonomous Hamiltonian System at Resonance 2:1:1

This paper presents an analysis of nonlinear oscillations of a near-autonomous two-degree-of-freedom Hamiltonian system, $2\pi$-periodic in time, in the neighborhood of a trivial equilibrium. It is assumed that in the autonomous case, for some set of parameters, the system experiences a multiple parametric resonance for which the frequencies of small linear oscillations in the neighborhood of the equilibrium are equal to two and one. It is also assumed that the Hamiltonian of perturbed motion contains only terms of even degrees with respect to perturbations, and its nonautonomous perturbing part depends on odd time harmonics. The analysis is performed in a small neighborhood of the resonance point of the parameter space. A series of canonical transformations is made to reduce the Hamiltonian of perturbed motion to a form whose main (model) part is characteristic of the resonance under consideration and the structure of nonautonomous terms. Regions of instability (regions of parametric resonance) of the trivial equilibrium are constructed analytically and graphically. A solution is presented to the problem of the existence and bifurcations of resonant periodic motions of the system which are analytic in fractional powers of a small parameter. As applications, resonant periodic motions of a double pendulum are constructed. The nearly constant lengths of the rods of the pendulum are prescribed periodic functions of time. The problem of the linear stability of these motions is solved.