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Home » Issues » Archive of Issues » 2019 - 2 issue » pp. 275-294

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Russia Moscow
Year
2019
Volume
29
Issue
2
Pages
275-294
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Section Mechanics
Title On multiple fourth-order resonances in a nonautonomous two-degree-of-freedom Hamiltonian system
Author(-s) Kholostova O.V.a
Affiliations Moscow Aviation Institutea
Abstract We consider the motion of a nonautonomous time-periodic two-degree-of-freedom Hamiltonian system in the vicinity of a trivial equilibrium being stable in the linear approximation. Fourth-order multiple (double or triple) resonance is assumed to be realized in the system. A list of all possible characteristic exponent sets corresponding to these resonant cases is given. Five qualitatively different approximate (model) Hamiltonian functions corresponding to these sets are obtained. For all cases of multiple resonances under study, sufficient conditions for the formal stability of the trivial equilibrium of the complete system are obtained, written as constraints on the coefficients of forms of the fourth degree in the normalized Hamiltonian functions of perturbed motion. A graphical interpretation of these conditions is given. The regions of formal stability are shown to be contained within the stability regions of each existing strong resonance considered separately, and the resonance coefficients corresponding to the weak resonances should take values from a limited range. Some questions of instability of the trivial equilibrium of the system in cases of multiple fourth-order resonances are considered. The found conditions of formal stability are examined at the points of multiple fourth-order resonances in the stability problem of cylindrical precession of a dynamically symmetric satellite-plate in the central Newtonian gravitational field on an elliptical orbit of arbitrary eccentricity.
Keywords Hamiltonian system, multiple fourth-order resonance, formal stability, satellite, cylindrical precession
UDC 531.36, 521.1
MSC 70H08, 70H12, 70H14, 70H15, 70M20
DOI 10.20537/vm190210
Received 10 June 2019
Language Russian
Citation Kholostova O.V. On multiple fourth-order resonances in a nonautonomous two-degree-of-freedom Hamiltonian system, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2019, vol. 29, issue 2, pp. 275-294.
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