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 Regul. Chaotic Dyn., 2017, Volume 22, Issue 7, Pages 773–781 (Mi rcd289)

On the Stability of Periodic Motions of an Autonomous Hamiltonian System in a Critical Case of the Fourth-order Resonance

Anatoly P. Markeev

Institute for Problems in Mechanics RAS, pr. Vernadskogo 101, str. 1, Moscow, 119526 Russia

Abstract: The problem of orbital stability of a periodic motion of an autonomous two-degreeof-freedom Hamiltonian system is studied. The linearized equations of perturbed motion always have two real multipliers equal to one, because of the autonomy and the Hamiltonian structure of the system. The other two multipliers are assumed to be complex conjugate numbers with absolute values equal to one, and the system has no resonances up to third order inclusive, but has a fourth-order resonance. It is believed that this case is the critical one for the resonance, when the solution of the stability problem requires considering terms higher than the fourth degree in the series expansion of the Hamiltonian of the perturbed motion.
Using Lyapunov’s methods and KAM theory, sufficient conditions for stability and instability are obtained, which are represented in the form of inequalities depending on the coefficients of series expansion of the Hamiltonian up to the sixth degree inclusive.

Keywords: Hamilton’s equations, stability, canonical transformations

 Funding Agency Grant Number Russian Science Foundation 14-21-00068 This research was supported by a grant from the Russian Science Foundation (14-21-00068) and was carried out at the Moscow Aviation Institute (National Research University).

DOI: https://doi.org/10.1134/S1560354717070012

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MSC: 70H05, 70H14, 70H15
Accepted:07.06.2017
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Citation: Anatoly P. Markeev, “On the Stability of Periodic Motions of an Autonomous Hamiltonian System in a Critical Case of the Fourth-order Resonance”, Regul. Chaotic Dyn., 22:7 (2017), 773–781

Citation in format AMSBIB
\Bibitem{Mar17} \by Anatoly P. Markeev \paper On the Stability of Periodic Motions of an Autonomous Hamiltonian System in a Critical Case of the Fourth-order Resonance \jour Regul. Chaotic Dyn. \yr 2017 \vol 22 \issue 7 \pages 773--781 \mathnet{http://mi.mathnet.ru/rcd289} \crossref{https://doi.org/10.1134/S1560354717070012} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000425980500001} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85042499594} 

• http://mi.mathnet.ru/eng/rcd289
• http://mi.mathnet.ru/eng/rcd/v22/i7/p773

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. B. S. Bardin, “On the stability of a periodic Hamiltonian system with one degree of freedom in a transcendental case”, Dokl. Math., 97:2 (2018), 161–163
2. A. P. Markeev, “Stability in the regular precession of an asymmetrical gyroscope in the critical case of fourth-order resonance”, Dokl. Phys., 63:7 (2018), 297–301