Almost sure and moment Lyapunov exponents for a stochastic beam equation
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.
stochastic partial differential equation
scientific article; zbMATH DE number 5151604