Multiple parametric resonance in Hamilton systems

✍ Scribed by A.P. Markeyev

• Publisher: Elsevier Science
• Year: 2006
• Tongue: English
• Weight: 550 KB
• Volume: 70
• Category: Article
• ISSN: 0021-8928
• DOI: 10.1016/j.jappmathmech.2006.06.001

✦ Synopsis

The stability of a linear Hamilton system, 2-periodic in time, with two degrees of freedom is investigated. The system depends on the parameters ␥ k (k = 1, 2, . . ., s) and . The parameter is assumed to be small. When = 0 the system is autonomous, and the roots of its characteristic equation are equal to ±i 1 and ±i 2 (i is the square root of -1 and 1 ≥0, 2 ≥ 0). Cases of multiple resonance are investigated when, for certain values of γ (0) k of the parameters ␥ k , the numbers 2 1 and 2 2 are simultaneously integers. All possible cases of such resonances are considered. For small but non-zero values of an algorithm for constructing regions of instability in the neighbourhood of resonance values of the parameters γ (0) k 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.