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 Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki: Year: Volume: Issue: Page: Find

 Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2016, Volume 26, Issue 3, Pages 418–438 (Mi vuu549)

MECHANICS

On the periodic motions of a Hamiltonian system in the neighborhood of unstable equilibrium in the presence of a double three-order resonance

A. I. Safonov, O. V. Kholostova

Department of Theoretical Mechanics, Moscow Aviation Institute (National Research University), Volokolamskoe shosse, 4, Moscow, 125993, Russia

Abstract: The paper considers the motion of a near-autonomous time-periodic two-degree-of-freedom Hamiltonian system in a neighborhood of trivial equilibrium being stable in the linear approximation. The third-order double resonance (fundamental and Raman) is assumed to occur in the system, at that Raman resonance can be strong or weak. In both cases the equilibrium considered is unstable in a full nonlinear system. Normalization of Hamiltonians of the perturbed motion is carried out in the terms up to the fourth order with respect to disturbance, taking into account the existing resonances. The problem of the existence and number of equilibrium positions of the corresponding approximate (model) systems is solved, and sufficient and necessary conditions for their stability are obtained. By Poincare's small parameter method, periodic motions of the initial full systems generated from the equilibrium positions of the model systems are constructed. The question of their stability in the linear approximation is solved. In particular, the conditions of the existence of stable (in the linear approximation) periodic motions in a small neighborhood of the unstable trivial equilibrium are obtained.

Keywords: Hamiltonian system, multiple resonance, stability, periodic motion.

 Funding Agency Grant Number Russian Foundation for Basic Research 14-01-00380

DOI: https://doi.org/10.20537/vm160310

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Bibliographic databases:

Document Type: Article
UDC: 531.011
MSC: 70H05, 70H14, 70H15, 70K45

Citation: A. I. Safonov, O. V. Kholostova, “On the periodic motions of a Hamiltonian system in the neighborhood of unstable equilibrium in the presence of a double three-order resonance”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 26:3 (2016), 418–438

Citation in format AMSBIB
\Bibitem{SafKho16} \by A.~I.~Safonov, O.~V.~Kholostova \paper On the periodic motions of a~Hamiltonian system in the neighborhood of unstable equilibrium in the presence of a~double three-order resonance \jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki \yr 2016 \vol 26 \issue 3 \pages 418--438 \mathnet{http://mi.mathnet.ru/vuu549} \crossref{https://doi.org/10.20537/vm160310} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3558451} \elib{http://elibrary.ru/item.asp?id=26726587}