Almost sure and moment Lyapunov exponents for a stochastic beam equation
type: Item
instance of: scholarly article
title: Almost sure and moment Lyapunov exponents for a stochastic beam equation
zbMATH Open document ID: 1123.60046
author: Marios Picas
author: Peter H. Baxendale
published in: Journal of Functional Analysis
publication date: 2007-05-10T00:00:00Z
review text: The transverse vibrations of an Euler-Bernoulli beam with axial tension P and axial white noise forcing are given by \[ my_{tt}+m\alpha_t+Eiy_{xxxx}-Py_{xx}=\sigma_{0y_{xx}}\dot W(t),\quad 0<x<\ell. \] With hinged endpoints \(y(0,t)=y_{xx}(0,t)=y(\ell,t)=y_{xx}(\ell,t)=0\) the solution may be written \[ y(x,t)=\sum^\infty_{n=1}\sin\left(\frac{n\pi x}{l}\right)c_n(t), \] where the amplitude \(c_n(t)\) of the \(n\)th mode of vibration is given by a two-dimensional linear stochastic (ordinary) differential equation. The main results give formulas for the almost sure and moment Lyapunov exponents for the SPDE in terms of the almost sure and moment Lyapunov exponents for each of the SDEs corresponding to modes which are present in the initial condition. It is shown that the almost sure and moment Lyapunov exponents for the SPDE depend sensitively on the initial distribution of energy amongst the infinitely many modes of vibration, and that they cannot be well approximated using only finitely many modes of vibration.
Mathematics Subject Classification ID: 60H15
zbMATH DE Number: 5151604
zbMATH Keywords: Euler-Bernoulli beam
zbMATH Keywords: stochastic partial differential equation
zbMATH Keywords: Lyapunov exponent
zbMATH Keywords: separation of variables
zbMATH Keywords: stochastic averaging
MaRDI profile type: Publication
full work available at URL: j.jfa.2006.10.006
OpenAlex ID: W1984608796
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DOI: 10.1016/J.JFA.2006.10.006
description: scientific article; zbMATH DE number 5151604