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Zh. Vychisl. Mat. Mat. Fiz., 2015, Volume 55, Number 1, Pages 56–73 (Mi zvmmf10135)  

This article is cited in 1 scientific paper (total in 1 paper)

Instability of a nonlinear system of two oscillators under main and combination resonances

N. A. Lyul'koab

a Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of Sciences, pr. Akademika Koptyuga 4, Novosibirsk, 630090, Russia
b Novosibirsk State University, ul. Pirogova 2, Novosibirsk, 630090, Russia

Abstract: A nonlinear reversible system of two oscillators depending on a small parameter $q>0$ is considered. The instability of the zero equilibrium of this system under a nonautonomous periodic perturbation is analyzed using the Krylov–Bogolyubov averaging method. In the case of main and combination resonances, independent integrals of the averaged autonomous nonlinear system are found, which are used to determine the maximum amplitude of oscillations of solutions to the original system for small $q$. In the case of the main resonance, the averaged system is reduced to a completely integrable Hamiltonian system by making a change of variables. In the case of combination resonance, the averaged system is integrated by applying the integrals found.

Key words: nonlinear system of two oscillators, parametric resonance, averaging method, first integrals, Hamiltonian systems.

Funding Agency Grant Number
Russian Academy of Sciences - Federal Agency for Scientific Organizations 15
Siberian Branch of Russian Academy of Sciences 30


DOI: https://doi.org/10.7868/S0044466915010160

Full text: PDF file (516 kB)
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English version:
Computational Mathematics and Mathematical Physics, 2015, 55:1, 53–70

Bibliographic databases:

UDC: 519.62
Received: 15.03.2013
Revised: 12.08.2014

Citation: N. A. Lyul'ko, “Instability of a nonlinear system of two oscillators under main and combination resonances”, Zh. Vychisl. Mat. Mat. Fiz., 55:1 (2015), 56–73; Comput. Math. Math. Phys., 55:1 (2015), 53–70

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. V. S. Belonosov, “Asymptotic analysis of the parametric instability of nonlinear hyperbolic equations”, Sb. Math., 208:8 (2017), 1088–1112  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
  • Журнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
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