Multiple parametric resonance in Hamilton systems

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Abstract

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.

Section snippets

Method of investigation

We will represent the equations of the boundaries of the stability and instability regions, adjacent to the point γk=γk(0) when ɛ = 0, in the form of the following seriesWhen solving specific problems, instead of the series (1.1) one must consider polynomials of finite degree in ɛ.

We substitute expansion (1.1) into the initial Hamiltonian F and expand it in series in powers of ɛ. The first term of this series F0 = F0(q1, q2, p1, p2) is equal to the functions F, calculated when ε=0,γk=γk(0). We will

The resonance ω1 = ω2 = 0

We will first consider the case when both frequencies of small oscillations are equal to zero when γk=γk(0). This case is encountered fairly rarely in practice. However, as will be seen from what follows, from the computational point of view many other special cases of multiple resonance reduce to it.

Depending on the value of the rank r of the matrix A0 of the equations of motion of the unperturbed system, when ω1 = ω2 = 0 we must distinguish four cases (r = 3, 2, 1 or 0), which do not reduce to one

The resonance 2ω1 = n1, ω2 = 0

We will consider multiple resonances when the frequency ω2 of small oscillations of the unperturbed system is equal to zero, while the other frequency ω1 is non-zero but 2ω1 = n1, where n1 is a natural number. Here we must distinguish four cases depending on the rank r of the matrix A0 of the equations of motion with unperturbed Hamiltonian F0 (it can be equal to three or two) and also depending on whether n1 is even or odd.

The resonance 2ω1 = 2 = n

We will now consider possible multiple resonances, when not one of the frequencies of small oscillations of the unperturbed system is equal to zero. In this section the frequencies are assumed to be equal: ω1 = ω2 = ω, where 2ω = n, and n is a natural number. Depending on the rank r of the matrix (A0  iωE) (it can be equal to three or two) here two cases are possible which do not reduce to one another.

The resonance 2ω1  =  n1, 2ω2 = n2 (n1  0, n2  0)

It remains to consider multiple resonances, when the frequencies of the unperturbed system are non-zero and different. Such resonances were investigated previously in Ref. 2. The normal form of the unperturbed Hamiltonian F0 is given by equality (4.4) in which σ1 = δ1ω1, σ2 = δ2ω2, δ1  =  ±1, δ2  =  ±1. After replacing the variables (4.5) we arrive at Hamiltonian (1.2) in which H0 = 0. Hence, the further investigation is carried as in Section 2.4.

It is useful to distinguish two cases in the calculations.

The stability of the steady rotation of a dynamically symmetrical satellite in an elliptic orbit

Suppose the centre of mass of the satellite moves in an elliptic orbit of eccentricity e in a central Newtonian gravitational field. The satellite is a rigid body, the central ellipsoid of which is a spheroid. The equatorial and polar moments of inertia of the satellite will be denoted by A and C. It is well known,10 that the problem of the motion of a satellite about a centre of mass under the action of gravitational moments allows of a particular solution, for which the axis of dynamic

Acknowledgement

This research was supported financially by the Russian Foundation for Basic Research (05-01-00386) and the programme for the support of the leading scientific schools (NSh-1477.2003.1).

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