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Regular and Chaotic Dynamics, 2020, Volume 25, Issue 1, Pages 111–120
DOI: https://doi.org/10.1134/S1560354720010098
(Mi rcd1052)
 

Special issue: In honor of Valery Kozlov for his 70th birthday

On Periodic Poincaré Motions in the Case of Degeneracy of an Unperturbed System

Anatoly P. Markeevab

a Moscow Aviation Institute (National Research University), Volokolamskoe sh. 4, Moscow, 125080 Russia
b Ishlinsky Institute for Problems in Mechanics RAS, pr. Vernadskogo 101-1, Moscow, 119526 Russia
Citations (1)
References:
Abstract: This paper is concerned with a one-degree-of-freedom system close to an integrable system. It is assumed that the Hamiltonian function of the system is analytic in all its arguments, its perturbing part is periodic in time, and the unperturbed Hamiltonian function is degenerate. The existence of periodic motions with a period divisible by the period of perturbation is shown by the Poincaré methods. An algorithm is presented for constructing them in the form of series (fractional degrees of a small parameter), which is implemented using classical perturbation theory based on the theory of canonical transformations of Hamiltonian systems. The problem of the stability of periodic motions is solved using the Lyapunov methods and KAM theory. The results obtained are applied to the problem of subharmonic oscillations of a pendulum placed on a moving platform in a homogeneous gravitational field. The platform rotates with constant angular velocity about a vertical passing through the suspension point of the pendulum, and simultaneously executes harmonic small-amplitude oscillations along the vertical. Families of subharmonic oscillations of the pendulum are shown and the problem of their Lyapunov stability is solved.
Keywords: Hamiltonian system, degeneracy, periodic motion, stability.
Funding agency Grant number
Ministry of Education and Science of the Russian Federation AAAA-A20-1200116900138-6
This research was carried out within the framework of the state assignment (registration No. AAAA-A20-1200116900138-6) at the Ishlinskii Institute for Problems in Mechanics, RAS, and at the Moscow Aviation Institute (National Research University).
Received: 04.09.2019
Accepted: 09.12.2019
Bibliographic databases:
Document Type: Article
MSC: 70H05, 70H12, 70H14
Language: English
Citation: Anatoly P. Markeev, “On Periodic Poincaré Motions in the Case of Degeneracy of an Unperturbed System”, Regul. Chaotic Dyn., 25:1 (2020), 111–120
Citation in format AMSBIB
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\by Anatoly P. Markeev
\paper On Periodic Poincaré Motions in the Case of Degeneracy of an Unperturbed System
\jour Regul. Chaotic Dyn.
\yr 2020
\vol 25
\issue 1
\pages 111--120
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\crossref{https://doi.org/10.1134/S1560354720010098}
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\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85079709334}
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  • https://www.mathnet.ru/eng/rcd/v25/i1/p111
  • This publication is cited in the following 1 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Abstract page:176
    References:52
     
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