The periodic motions of a non-autonomous Hamiltonian system with two degrees of freedom at parametric resonance of the fundamental type

Abstract

The motion of an almost autonomous Hamiltonian system with two degrees of freedom, 2π-periodic in time, is considered. It is assumed that the origin is an equilibrium position of the system, the linearized unperturbed system is stable, and its characteristic exponents ± iωj ( j = 1,2) are pure imaginary. In addition, it is assumed that the number 2 ω 1 is approximately an integer, that is, the system exhibits parametric resonance of the fundamental type. Using Poincaré's theory of periodic motion and KAM-theory, it is shown that 4π-periodic motions of the system exist in a fairly small neighbourhood of the origin, and their bifurcation and stability are investigated. As applications, periodic motions are constructed in cases of parametric resonance of the fundamental type in the following problems: the plane elliptical restricted three-body problem near triangular libration points, and the problem of the motion of a dynamically symmetrical artificial satellite near its cylindrical precession in an elliptical orbit of small eccentricity.