Abstract:
We consider the problem on parametric resonance for linear periodic Hamiltonian systems depending on a small parameter. We propose new formulae based on the methods of the perturbation theory for linear operators in the problem on approximate construction of multipliers for linear non-autonomous periodic Hamiltonian systems. We focus on obtaining the formulae for the first correctors of perturbations of multiple definite and indefinite multipliers. The proposed formulae lead to new Lyapunov stability criteria for linear periodic Hamiltonian systems in critical cases. We consider applications to the problem on parametric resonance in main resonances. The obtained results are formulated in terms of the original equations and lead us to effective formulae and algorithms. The effectiveness of the proposed formulae is demonstrated by solving the problem of plotting the boundaries of the stability regions of triangular libration points of a planar bounded elliptic three-body problem.
The research of the third author is made in the framework of State Task of the Ministry of Science and
Higher Education of Russian Federation (code of scientific theme FZWU-2020-0027).
Received: 18.02.2021
Bibliographic databases:
Document Type:
Article
UDC:517.958
Language: English
Original paper language: Russian
Citation:
M. G. Yumagulov, L. S. Ibragimova, A. S. Belova, “Perturbation theory methods in problem of parametric resonance for linear periodic Hamiltonian systems”, Ufa Math. J., 13:3 (2021), 174–190
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\paper Perturbation theory methods in problem of parametric resonance for linear periodic Hamiltonian systems
\jour Ufa Math. J.
\yr 2021
\vol 13
\issue 3
\pages 174--190
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Linking options:
https://www.mathnet.ru/eng/ufa584
https://doi.org/10.13108/2021-13-3-174
https://www.mathnet.ru/eng/ufa/v13/i3/p178
This publication is cited in the following 3 articles:
S. V. Akmanova, M. G. Yumagulov, “O lokalnykh bifurkatsiyakh v nelineinykh nepreryvno-diskretnykh dinamicheskikh sistemakh”, Izv. vuzov. Matem., 2025, no. 2, 3–14
M. G. Yumagulov, L. S. Ibragimova, “Equivalent Differential Equations in Problems of Control Theory
and the Theory of Hamiltonian Systems”, Diff Equat, 60:1 (2024), 23
A. S. Belova, “Stability of Equilibrium Points for a Hamiltonian Systems with Two Degrees of Freedom in the Problem of Parametric Resonance”, Lobachevskii J Math, 43:6 (2022), 1486
Citation in format AMSBIB
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