Nonuniform exponential contractions and Lyapunov sequences
type: Item
instance of: scholarly article
title: Nonuniform exponential contractions and Lyapunov sequences
zbMATH Open document ID: 1173.37029
author: Luís Barreira
author: Claudia Valls
published in: Journal of Differential Equations
publication date: 2009-06-10T00:00:00Z
review text: The paper deals with a nonautonomous dynamics with discrete time obtained from the product of linear operators. For this dynamics, it is shown that a nonuniform exponential contraction can be completely characterized in terms of so-called strict Lyapunov sequences. Nonuniform exponential contractions include as a very particular case the uniform exponential contractions that correspond to have a uniform asymptotic stability of the dynamics. The paper also presents ``inverse theorems'' that give explicitly strict Lyapunov sequences for each nonuniform exponential contraction. Essentially, the Lyapunov sequences are obtained in terms of what are usually called Lyapunov norms, that is, norms with respect to which the behavior of a nonuniform exponential contraction becomes uniform. It is also shown how the characterization of nonuniform exponential contractions in terms of quadratic Lyapunov sequences can be used to establish in a simple manner the persistence of the asymptotic stability of a nonuniform exponential contraction under sufficiently small linear or nonlinear perturbations. In addition, an appropriate version of the results obtained is described in the context of ergodic theory showing that the existence of an eventually strict Lyapunov function implies that all Lyapunov exponents are negative almost everywhere.
Mathematics Subject Classification ID: 37D25
Mathematics Subject Classification ID: 93D30
zbMATH DE Number: 5563805
zbMATH Keywords: Lyapunov sequences
zbMATH Keywords: nonuniform exponential contractions
zbMATH Keywords: nonautonomous dynamics
zbMATH Keywords: Lyapunov norms
MaRDI profile type: Publication
full work available at URL: j.jde.2009.01.020
OpenAlex ID: W2024430905
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DOI: 10.1016/J.JDE.2009.01.020
description: scientific article; zbMATH DE number 5563805