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Sib. Èlektron. Mat. Izv., 2014, Volume 11, Pages 675–694 (Mi semr514)  

This article is cited in 2 scientific papers (total in 2 papers)

Computational mathematics

Asymptotic and numerical analysis of parametric resonance in a nonlinear system of two oscillators

N. A. Lyulkoab, N. A. Kudryavtsevab, A. N. Kudryavtsevcb

a Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk
b Novosibirsk State University
c Khristianovich Institute of Theoretical and Applied Mechanics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk

Abstract: A parametric resonance in a nonlinear system of ordinary differential equations, which is a mathemetical model of a water–oil gas containing layer, is considered. The Krylov–Bolgoliubov–Mitropolsky averaging method is applied to investigate the instability of zero solution of the system and deduce averaged equations for time evolution of the amplitude of oscillations in the cases of main and combinational resonances. The original and averaged equations are also integrated numerically with a high-order strong stability preserving Runge–Kutta scheme. By comparing the numerical solutions it is shown that the averaged equations enable us to predict correctly the maximum amplitude of oscillations and the time moment when it is achieved. The dependence of resonance characteritics on the small parameter is also studied.

Keywords: instability in nonlinear system of two oscillators, main and combinational parametric resonances, asymptotic and numerical analysis of resonance.

Full text: PDF file (1099 kB)
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UDC: 517.928
MSC: 34C15,34C29
Received April 21, 2014, published August 30, 2014

Citation: N. A. Lyulko, N. A. Kudryavtseva, A. N. Kudryavtsev, “Asymptotic and numerical analysis of parametric resonance in a nonlinear system of two oscillators”, Sib. Èlektron. Mat. Izv., 11 (2014), 675–694

Citation in format AMSBIB
\Bibitem{LyuKudKud14}
\by N.~A.~Lyulko, N.~A.~Kudryavtseva, A.~N.~Kudryavtsev
\paper Asymptotic and numerical analysis of parametric resonance in a nonlinear system of two oscillators
\jour Sib. \`Elektron. Mat. Izv.
\yr 2014
\vol 11
\pages 675--694
\mathnet{http://mi.mathnet.ru/semr514}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. N. A. Lyul'ko, “Instability of a nonlinear system of two oscillators under main and combination resonances”, Comput. Math. Math. Phys., 55:1 (2015), 53–70  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    2. 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
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