Stability in the almost everywhere sense: a linear transfer operator approach. (English) Zbl 1201.34085
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.
Reviewer: Yuri V. Rogovchenko (Umeå)
MSC:
34D20 | Stability of solutions to ordinary differential equations |
34D45 | Attractors of solutions to ordinary differential equations |
37B25 | Stability of topological dynamical systems |
65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
dynamical system; almost everywhere stability; linear transfer operator; advection equation; finite element method; Perron-Frobenius operatorReferences:
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