Some conditions for the existence and stability of periodic oscillations in non-linear non-autonomous hamiltonian systems

Abstract

The sufficient conditions for the existence and uniqueness of periodic solutions are obtained for non-autonomous Hamiltonian systems by the method of continuation with respect to the parameter /1/ (similar results were established for certain vector equations by other methods in /2, 3/). Using the theorem on the directed width of stability regions /4/, stability criteria to a first approximation of these solutions are obtained. The effect of small dissipative forces on stability is investigated. Systems are considered in which some of the generalized coordinates are angular. The conditions for the existence, uniqueness, and stability are obtained, as well as the upper bounds of solutions that correspond to periodic rotational motions of the angular coordinates with any preassigned average velocities that are multiples of the perturbing effect. The periodic oscillatory and rotational motions of two coupled pendulums are considered, as an example.