Multiple parametric resonance in Hamilton systems

The stability of a linear Hamilton system, 2π-periodic in time, with two degrees of freedom is investigated. The system depends on the parameters γk(k = 1, 2, …, s) and ɛ. The parameter ɛ is assumed to be small. When ɛ = 0 the system is autonomous, and the roots of its characteristic equation are equal to ±iω1 and ±iω2 (i is the square root of −1 and ω1 ≥0, ω2 ≥ 0). Cases of multiple resonance are investigated when, for certain values of γk(0) of the parameters γk, the numbers 2ω1 and 2ω2 are simultaneously integers. All possible cases of such resonances are considered. For small but non-zero values of ɛ an algorithm for constructing regions of instability in the neighbourhood of resonance values of the parameters γk(0) is proposed. Using this algorithm, the linear problem of the stability of the steady rotation of a dynamically symmetrical satellite when there are multiple resonances is investigated. The orbit of the centre of mass is assumed to be elliptical, the eccentricity of the orbit is small, and in the unperturbed motion the axis of symmetry of the satellite is perpendicular to the orbital plane.