Stability in the almost everywhere sense: a linear transfer operator approach
type: Item
instance of: scholarly article
title: Stability in the almost everywhere sense: a linear transfer operator approach
zbMATH Open document ID: 1201.34085
author: B. E. Eshmatov
published in: Journal of Mathematical Analysis and Applications
publication date: 2010-05-19T00:00:00Z
review text: The authors study everywhere uniform stability problem for a nonlinear differential equation \(\dot{x}=f\left( x\right) \) on a compact phase space \(X\subset\mathbb{R}^{n}\) using the associated advection equation \[ \frac{\partial\rho}{\partial t}=-\nabla\cdot\left( \rho f\right) . \] Connection between the stability of an ordinary differential equation and a linear partial differential equation provides new computational tools for the analysis of nonlinear dynamical systems. The principal result in the paper states that almost everywhere uniform stability of an attractor set for a continuous dynamical system is equivalent to existence of a positive solution, termed ``Lyapunov density,'' to an advection type partial differential equation. A finite element based numerical method is used for computing the density in lower dimensions.
Mathematics Subject Classification ID: 34D20
Mathematics Subject Classification ID: 34D45
Mathematics Subject Classification ID: 37B25
Mathematics Subject Classification ID: 65M60
zbMATH DE Number: 5710075
zbMATH Keywords: dynamical system
zbMATH Keywords: almost everywhere stability
zbMATH Keywords: linear transfer operator
zbMATH Keywords: advection equation
zbMATH Keywords: finite element method
zbMATH Keywords: Perron-Frobenius operator
reviewed by: Yuriy V. Rogovchenko
MaRDI profile type: Publication
full work available at URL: j.jmaa.2010.02.032
OpenAlex ID: W2069876442
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DOI: 10.1016/J.JMAA.2010.02.032
description: scientific article; zbMATH DE number 5710075