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Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva, 2021, Volume 23, Number 3, Pages 285–294
DOI: https://doi.org/10.15507/2079-6900.23.202103.285-294
(Mi svmo801)
 

Mathematics

On maintaining the stability of the equilibrium of nonlinear oscillators under conservative perturbations

A. A. Kosov

Matrosov Institute for System Dynamics and Control Theory of Siberian Branch of Russian Academy of Sciences, Irkutsk
References:
Abstract: The problem of Yu.N. Bibikov on maintaining the stability of the equilibrium position of two interconnected nonlinear oscillators under the action of small, in a certain sense, conservative perturbing forces is considered. With different methods of reducing the system to the Hamiltonian form, some features are revealed for the case when the perturbing forces of the interaction of two oscillators are potential. The conditions for preserving the stability and instability of the equilibrium of two oscillators for the case of sufficiently small disturbing forces are obtained. The problem of maintaining the stability of the equilibrium under conservative perturbations is also considered in the more general situation of an arbitrary number of oscillators with power potentials with rational exponents, which leads to the case of a generalized homogeneous potential of an unperturbed system. The example given shows the applicability of the proposed approach in the case when the order of smallness of the perturbing forces coincides with the order of smallness of the unperturbed Hamiltonian.
Keywords: nonlinear oscillators, Hamiltonian system, stability.
Funding agency Grant number
Russian Foundation for Basic Research 19-08-00746
Document Type: Article
UDC: 517.925
MSC: Primary 34D20; Secondary 34H15, 70E50
Language: Russian
Citation: A. A. Kosov, “On maintaining the stability of the equilibrium of nonlinear oscillators under conservative perturbations”, Zhurnal SVMO, 23:3 (2021), 285–294
Citation in format AMSBIB
\Bibitem{Kos21}
\by A.~A.~Kosov
\paper On maintaining the stability of the equilibrium of nonlinear oscillators under conservative perturbations
\jour Zhurnal SVMO
\yr 2021
\vol 23
\issue 3
\pages 285--294
\mathnet{http://mi.mathnet.ru/svmo801}
\crossref{https://doi.org/10.15507/2079-6900.23.202103.285-294}
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