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Vestnik YuUrGU. Ser. Mat. Model. Progr., 2012, Issue 14, Pages 39–52 (Mi vyuru80)  

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

Mathematical Modelling

Asymptotic Stability of Solutions to One Class of Nonlinear Second-Order Differential Equations with Parameters

G. V. Demidenkoab, K. M. Dulinab, I. I. Matveevaab

a Sobolev Institute of Mathematics
b Novosibirsk State University (Novosibirsk, Russian Federation)

Abstract: We consider a class of nonlinear second-order ordinary differential equations with parameters. Differential equations of such type arise when studying oscillations of an «inversed pendulum» in which the pivot point vibrates periodically. We establish conditions under which the zero solution is asymptotically stable. We obtain estimates for the attraction domain of the zero solution and establish estimates for the decay rate of solutions at infinity. Obtaining the results, we use a criterion for asymptotic stability of the zero solution to systems of linear ordinary differential equations with periodic coefficients. The criterion is formulated in terms of solvability of a special boundary value problem for the Lyapunov differential equation on the interval. The estimates of the attraction domain of the zero solution and estimates for the decay rate of the solutions at infinity are established by the use of the norm of the solution to the boundary value problem.

Keywords: second-order differential equations, periodic coefficients, asymptotic stability, Lyapunov differential equation.

Full text: PDF file (198 kB)
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UDC: 517.925.44
MSC: 34K20
Received: 17.07.2012

Citation: G. V. Demidenko, K. M. Dulina, I. I. Matveeva, “Asymptotic Stability of Solutions to One Class of Nonlinear Second-Order Differential Equations with Parameters”, Vestnik YuUrGU. Ser. Mat. Model. Progr., 2012, no. 14, 39–52

Citation in format AMSBIB
\Bibitem{DemDulMat12}
\by G.~V.~Demidenko, K.~M.~Dulina, I.~I.~Matveeva
\paper Asymptotic Stability of Solutions to One Class of Nonlinear Second-Order Differential Equations with Parameters
\jour Vestnik YuUrGU. Ser. Mat. Model. Progr.
\yr 2012
\issue 14
\pages 39--52
\mathnet{http://mi.mathnet.ru/vyuru80}


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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. G. V. Demidenko, A. V. Dulepova, “On stability of the inverted pendulum motion with a vibrating suspension point”, J. Appl. Industr. Math., 12:4 (2018), 607–618  mathnet  crossref  crossref
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