Lyapunov functions for trichotomies with growth rates
type: Item
instance of: scholarly article
title: Lyapunov functions for trichotomies with growth rates
zbMATH Open document ID: 1197.34094
author: Luís Barreira
author: Claudia Valls
published in: Journal of Differential Equations
publication date: 2009-12-22T00:00:00Z
review text: Let \(T(t, \tau)\) be the linear evolution operator associated to the linear equation \[ x' = A(t)x \] in the finite-dimensional space \({\mathbb R}^n\). We say that \(x' = A(t)x\) admits a \(\rho\)-nonuniform exponential contraction if there exist constants \(d\), \(D > 0\) and \(\varepsilon \geq 0\) such that \(\| T(t, \tau) \| \leq D e^{-d(\rho(t) - \rho(\tau)) + \varepsilon|\rho(\tau)|}\) for \(t \geq \tau > 0\). When the equation admits a \(\rho\)-nonuniform exponential contraction with \(\varepsilon = 0\), we say that it admits a \(\rho\)-uniform exponential contraction. In this paper, in strong contrast with the usual stable, unstable, and central behavior, the authors consider asymptotic rates of the form \(e^{c \rho(t)}\) determined by an arbitrary function \(\rho(t)\). The usual notion of nonuniform exponential contraction corresponds to the particular case when \(\rho(t) = t\), that is, \(\| T(t, \tau) \| \leq D e^{-d(t - \tau) + \varepsilon \tau}\) for \(t \geq \tau > 0\) [see \textit{L. Barreira} and \textit{C. Valls}, J. Differ. Equations 217, No.~1, 204--248 (2005; Zbl 1088.34053)]. The notion of nonuniform polynomial contraction is recovered when \(\rho(t) = \log (1 + t)\), that is, \[ \| T(t, \tau) \| \leq D \left( \frac{1 + t}{1 + \tau} \right)^{-d} (1 + \tau)^\varepsilon \] for \(t \geq \tau > 0\) [see \textit{L. Barreira} and \textit{C. Valls}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71, No.~11, 5208--5219 (2009; Zbl 1181.34046)]. Let \(V : {\mathbb R}_0^+ \times {\mathbb R}^n \to {\mathbb R}_0^-\) be a function, and assume that there exist \(C > 0\) and \(\delta \geq 0\) such that \(|V(t, x)| \leq C e^{\delta |\rho(t)|} \| x \|\) for every \(t \geq 0\) and \(x \in {\mathbb R}^n\). Given \(\alpha > 0\) and \(\gamma \geq 0\), we say that \(V\) is a strict Lyapunov function for the equation \(x' = A(t)x\) if \[ V(t, T(t, \tau) x) \leq \alpha^{\rho(t) - \rho(\tau)} V(\tau, x) \] for \(t \geq \tau\), and \[ V(\tau, x) \geq \frac…
reviewed by: Kazuhiro Sakai
Mathematics Subject Classification ID: 34D09
Mathematics Subject Classification ID: 34D20
Mathematics Subject Classification ID: 34D08
zbMATH DE Number: 5651186
zbMATH Keywords: Lyapunov functions
zbMATH Keywords: nonuniform exponential trichotomy
MaRDI profile type: Publication
full work available at URL: j.jde.2009.07.001
OpenAlex ID: W1995459866
cites work: Q5431024
cites work: Stability of nonautonomous differential equations in Hilbert spaces
cites work: Growth rates and nonuniform hyperbolicity
cites work: Stability of nonautonomous differential equations
cites work: Polynomial growth rates
cites work: PARTIALLY HYPERBOLIC DYNAMICAL SYSTEMS
cites work: Center manifolds for infinite dimensional nonautonomous differential equations
cites work: Q4258125
cites work: Existence and roughness of the exponential dichotomy for skew-product semiflow in Banach spaces
cites work: Center manifolds for invariant sets
cites work: Center manifolds for smooth invariant manifolds
cites work: Asymptotic stability without uniform stability: Almost periodic coefficients
cites work: Dichotomies and reducibility
cites work: Dichotomies in stability theory
cites work: Q4774403
cites work: Exponential trichotomy of differential systems
cites work: Almost reducibility of linear quasi-periodic systems
cites work: Q3847146
cites work: Q4039920
cites work: Geometric theory of semilinear parabolic equations
cites work: The stable, center-stable, center, center-unstable, unstable manifolds
cites work: Linear differential equations and functional analysis. I
cites work: Q5599358
cites work: A reduction principle for nonautonomous systems in infinite-dimensional spaces
cites work: Admissible perturbations of exponential dichotomy roughness
cites work: Q5646734
cites work: FAMILIES OF INVARIANT MANIFOLDS CORRESPONDING TO NONZERO CHARACTERISTIC EXPONENTS
cites work: CHARACTERISTIC LYAPUNOV EXPONENTS AND SMOOTH ERGODIC THEORY
cites work: Q5340295
cites work: Robustness of exponential dichotomies in infinite-dimensional dynamical systems
cites work: Exponential dichotomy roughness on Banach spaces
cites work: Existence of dichotomies and invariant splittings for linear differential systems. I
cites work: Existence of dichotomies and invariant splittings for linear differential systems. III
cites work: Dynamics of evolutionary equations
cites work: Q3833289
cites work: Q4012892
cites work: Center manifolds and contractions on a scale of Banach spaces
DOI: 10.1016/J.JDE.2009.07.001
description: scientific article; zbMATH DE number 5651186