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 Regul. Chaotic Dyn., 2015, Volume 20, Issue 6, Pages 627–648 (Mi rcd33)

On the Stability of Periodic Hamiltonian Systems with One Degree of Freedom in the Case of Degeneracy

Boris S. Bardina, Victor Lancharesb

a Department of Theoretical Mechanics, Faculty of Applied Mathematics and Physics, Moscow Aviation Institute, Volokolamskoe sh. 4, Moscow, 125993 Russia
b Departamento de Matemáticas y Computación, CIME, Universidad de La Rioja, 26004 Logroño, Spain

Abstract: We deal with the stability problem of an equilibrium position of a periodic Hamiltonian system with one degree of freedom. We suppose the Hamiltonian is analytic in a small neighborhood of the equilibrium position, and the characteristic exponents of the linearized system have zero real part, i.e., a nonlinear analysis is necessary to study the stability in the sense of Lyapunov. In general, the stability character of the equilibrium depends on nonzero terms of the lowest order $N$ $(N>2)$ in the Hamiltonian normal form, and the stability problem can be solved by using known criteria.
We study the so-called degenerate cases, when terms of order higher than $N$ must be taken into account to solve the stability problem. For such degenerate cases, we establish general conditions for stability and instability. Besides, we apply these results to obtain new stability criteria for the cases of degeneracy, which appear in the presence of first, second, third and fourth order resonances.

Keywords: Hamiltonian systems, Lyapunov stability, stability theory, normal forms, KAM theory, Chetaev's function, resonance

 Funding Agency Grant Number Russian Science Foundation 14-21-00068 Ministry of Science and Innovation of Spanish MTM2011-28227-C0MTM2014-59433-C2-2-P The first author acknowledges financial support from the Russian Scientific Foundation (project No.14-21-00068 at the Moscow Aviation Institute (National Research University)). The second author acknowledges financial support from the Spanish Ministry of Science and Innovation (projects MTM2011-28227-C0 and MTM2014-59433-C2-2-P).

DOI: https://doi.org/10.1134/S1560354715060015

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Bibliographic databases:

MSC: 34D20, 37C75, 37J4
Accepted:05.10.2015
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Citation: Boris S. Bardin, Victor Lanchares, “On the Stability of Periodic Hamiltonian Systems with One Degree of Freedom in the Case of Degeneracy”, Regul. Chaotic Dyn., 20:6 (2015), 627–648

Citation in format AMSBIB
\Bibitem{BarLan15} \by Boris~S.~Bardin, Victor Lanchares \paper On the Stability of Periodic Hamiltonian Systems with One Degree of Freedom in the Case of Degeneracy \jour Regul. Chaotic Dyn. \yr 2015 \vol 20 \issue 6 \pages 627--648 \mathnet{http://mi.mathnet.ru/rcd33} \crossref{https://doi.org/10.1134/S1560354715060015} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3431180} \adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2015RCD....20..627B} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000365809000001} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84948964011} 

• http://mi.mathnet.ru/eng/rcd33
• http://mi.mathnet.ru/eng/rcd/v20/i6/p627

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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. Rodrigo Gutierrez, Claudio Vidal, “Stability of Equilibrium Points for a Hamiltonian Systems with One Degree of Freedom in One Degenerate Case”, Regul. Chaotic Dyn., 22:7 (2017), 880–892
2. 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
3. N. Xue, X. Li, “The linearization of periodic Hamiltonian systems with one degree of freedom under the Diophantine condition”, J. Differ. Equ., 264:2 (2018), 604–623
4. Boris S. Bardin, Víctor Lanchares, “Stability of a One-degree-of-freedom Canonical System in the Case of Zero Quadratic and Cubic Part of a Hamiltonian”, Regul. Chaotic Dyn., 25:3 (2020), 237–249