Stability properties of solutions of linear second order differential equations with random coefficients
type: Item
instance of: scholarly article
title: Stability properties of solutions of linear second order differential equations with random coefficients
zbMATH Open document ID: 1189.34106
author: László Hatvani
author: Sándor Csörgő
published in: Journal of Differential Equations
publication date: 2009-12-22T00:00:00Z
review text: The starting point is the non-autonomous equation \[ x^{\prime\prime}+a^2(t)x=0,\;\; t\geq 0 \] in the hypotheses that the coefficient function \(a^2\) is not a deterministic periodic function, and that the lengths of the time intervals between consecutive jump points are independent, positive and not necessarily identically distributed random variables with values in the interval \([0,2T]\). The authors give sufficient conditions for both stability and instability of the equilibrium position \(x=0\) in the most general case, and hence, they extend some results given in [\textit{L. Hatvani} and \textit{L. Stachó}, Arch. Math., Brno 34, No.~1, 119--126 (1998; Zbl 0915.34051)] or in others L. Hatvani's papers. The main results are given in Theorem 2.3, Theorem 2.4, Theorem 2.5, Theorem 2.10 and Theorem 2.11. We remark the complex and very explicitly proofs of these results. More examples and more remarks, which explain the possible applications of this equation type, continue the very important theoretical results. Some numerical results are presented.
reviewed by: Romeo Negrea
Mathematics Subject Classification ID: 34F05
Mathematics Subject Classification ID: 34C11
Mathematics Subject Classification ID: 70L05
zbMATH DE Number: 5651180
zbMATH Keywords: small solutions
zbMATH Keywords: random oscillations
zbMATH Keywords: parametric resonance
zbMATH Keywords: stochastic parametric resonance
zbMATH Keywords: problem of swinging
MaRDI profile type: Publication
full work available at URL: j.jde.2009.08.001
OpenAlex ID: W2015690214
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DOI: 10.1016/J.JDE.2009.08.001
description: scientific article; zbMATH DE number 5651180