NONUNIFORM EXPONENTIAL BEHAVIOUR AND TOPOLOGICAL EQUIVALENCE

2015 ◽  
Vol 58 (2) ◽  
pp. 279-291
Author(s):  
LUIS BARREIRA ◽  
LIVIU HORIA POPESCU ◽  
CLAUDIA VALLS
Keyword(s):  
Asymptotic Behaviour ◽  
Ergodic Theory ◽  
Canonical Form ◽  
Exponential Dichotomy ◽  
Topological Equivalence ◽  
Exponential Dichotomies ◽  
Exponential Behaviour ◽  
Nonuniform Exponential Dichotomies ◽  
Nonuniform Exponential Dichotomy ◽  
Topological Conjugacies

AbstractWe show that any evolution family with a strong nonuniform exponential dichotomy can always be transformed by a topological equivalence to a canonical form that contracts and/or expands the same in all directions. We emphasize that strong nonuniform exponential dichotomies are ubiquitous in the context of ergodic theory. The main novelty of our work is that we are able to control the asymptotic behaviour of the topological conjugacies at the origin and at infinity.

scholarly journals Permanence of nonuniform nonautonomous hyperbolicity for infinite-dimensional differential equations

2021 ◽  
pp. 1-27
Author(s):  
Tomás Caraballo ◽  
Alexandre N. Carvalho ◽  
José A. Langa ◽  
Alexandre N. Oliveira-Sousa
Keyword(s):  
Differential Equations ◽  
Continuous Dependence ◽  
Exponential Dichotomy ◽  
Nonlinear Evolution ◽  
Nonuniform Hyperbolicity ◽  
Exponential Dichotomies ◽  
Infinite Dimensional ◽  
Linear Evolution ◽  
Nonuniform Exponential Dichotomies ◽  
Nonuniform Exponential Dichotomy

In this paper, we study stability properties of nonuniform hyperbolicity for evolution processes associated with differential equations in Banach spaces. We prove a robustness result of nonuniform hyperbolicity for linear evolution processes, that is, we show that the property of admitting a nonuniform exponential dichotomy is stable under perturbation. Moreover, we provide conditions to obtain uniqueness and continuous dependence of projections associated with nonuniform exponential dichotomies. We also present an example of evolution process in a Banach space that admits nonuniform exponential dichotomy and study the permanence of the nonuniform hyperbolicity under perturbation. Finally, we prove persistence of nonuniform hyperbolic solutions for nonlinear evolution processes under perturbations.

Smooth Dependence of Topological Conjugacies on Parameters

2011 ◽  
Vol 11 (2) ◽  
Author(s):  
Luis Barreira ◽  
Claudia Valls
Keyword(s):  
Exponential Dichotomy ◽  
Linear Equations ◽  
Smooth Dependence ◽  
Nonuniform Exponential Dichotomy ◽  
Topological Conjugacies

AbstractFor nonautonomous linear equations xʹ = A(t)x with a nonuniform exponential dichotomy, we show that under sufficiently small C

Exponential dichotomies with respect to a sequence of norms and admissibility

2014 ◽  
Vol 25 (03) ◽  
pp. 1450024 ◽  
Author(s):  
Luis Barreira ◽  
Davor Dragičević ◽  
Claudia Valls
Keyword(s):  
Exponential Dichotomy ◽  
Linear Operators ◽  
Conditional Stability ◽  
Exponential Behavior ◽  
Exponential Dichotomies ◽  
Linear Perturbations ◽  
Nonuniform Exponential Dichotomy

For a nonautonomous dynamics defined by a sequence of linear operators, we introduce the notion of an exponential dichotomy with respect to a sequence of norms and we characterize it completely in terms of the admissibility in lp spaces, both for the space of perturbations and the space of solutions. This allows unifying the notions of uniform and nonuniform exponential behavior. Moreover, we consider the general case of a noninvertible dynamics. As a nontrivial application we show that the conditional stability of a nonuniform exponential dichotomy persists under sufficiently small linear perturbations.

Strong nonuniform spectrum for arbitrary growth rates

2017 ◽  
Vol 19 (02) ◽  
pp. 1650008 ◽  
Author(s):  
Luis Barreira ◽  
Davor Dragičević ◽  
Claudia Valls
Keyword(s):  
Growth Rates ◽  
Exponential Dichotomy ◽  
Complete Characterization ◽  
Initial Time ◽  
Exponential Dependence ◽  
Conditional Stability ◽  
Exponential Dichotomies ◽  
Exponential Rates ◽  
Nonuniform Exponential Dichotomies

We consider the notion of strong nonuniform spectrum for a nonautonomous dynamics with discrete time obtained from a sequence of matrices, which is defined in terms of the existence of strong nonuniform exponential dichotomies with an arbitrarily small nonuniform part. The latter exponential dichotomies are ubiquitous in the context of ergodic theory and correspond to have both lower and upper bounds along the stable and unstable directions, besides possibly a nonuniform conditional stability although with an arbitrarily small exponential dependence on the initial time. Moreover, we consider arbitrary growth rates instead of only the usual exponential rates. We give a complete characterization of the possible strong nonuniform spectra and for a Lyapunov regular trajectory, we show that the spectrum is the set of Lyapunov exponents. In addition, we provide explicit examples of nonautonomous dynamics for all possible strong nonuniform spectra. A remarkable consequence of our results is that for a sequence of matrices [Formula: see text], either [Formula: see text] does not admit a strong exponential dichotomy for any [Formula: see text], or if [Formula: see text] admits an exponential dichotomy for some [Formula: see text], then it also admits a strong exponential dichotomy for that [Formula: see text]. We emphasize that this result is not in the literature even in the special case of uniform exponential dichotomies.

scholarly journals Smooth Stable Manifold for Delay Differential Equations with Arbitrary Growth Rate

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 105
Author(s):  
Lokesh Singh ◽  
Dhirendra Bahuguna
Keyword(s):  
Differential Equation ◽  
Growth Rate ◽  
Invariant Manifold ◽  
Delay Differential Equation ◽  
Delay Differential Equations ◽  
Stable Manifold ◽  
Exponential Dichotomy ◽  
Delay Differential ◽  
Stable Invariant ◽  
Nonuniform Exponential Dichotomy

In this article, we construct a C1 stable invariant manifold for the delay differential equation x′=Ax(t)+Lxt+f(t,xt) assuming the ρ-nonuniform exponential dichotomy for the corresponding solution operator. We also assume that the C1 perturbation, f(t,xt), and its derivative are sufficiently small and satisfy smoothness conditions. To obtain the invariant manifold, we follow the method developed by Lyapunov and Perron. We also show the dependence of invariant manifold on the perturbation f(t,xt).

scholarly journals Nonuniform Exponential Dichotomies in Terms of Lyapunov Functions for Noninvertible Linear Discrete-Time Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Ioan-Lucian Popa ◽  
Mihail Megan ◽  
Traian Ceauşu
Keyword(s):  
Discrete Time ◽  
Lyapunov Functions ◽  
Exponential Dichotomies ◽  
Discrete Time Systems ◽  
Nonuniform Exponential Dichotomies ◽  
Time Systems

The aim of this paper is to give characterizations in terms of Lyapunov functions for nonuniform exponential dichotomies of nonautonomous and noninvertible discrete-time systems.

On the exponential behaviour of non-autonomous difference equations

2013 ◽  
Vol 56 (3) ◽  
pp. 643-656
Author(s):  
Luis Barreira ◽  
Claudia Valls
Keyword(s):  
Asymptotic Behaviour ◽  
Lyapunov Exponents ◽  
Difference Equations ◽  
Small Perturbations ◽  
Exponential Behaviour ◽  
Exponential Rates

AbstractGiven a sequence of matrices(Am)m∈ℕwhose Lyapunov exponents are limits, we show that this asymptotic behaviour is reproduced by the sequencesxm+1=Amxm+ fm(xm)for any sufficiently small perturbationsfm. We also consider the general case of exponential rates ecρmfor an arbitrary increasing sequenceρm. Our approach is based on Lyapunov's theory of regularity.

Tempered exponential dichotomies and Lyapunov exponents for perturbations

2016 ◽  
Vol 18 (05) ◽  
pp. 1550058 ◽  
Author(s):  
Luis Barreira ◽  
Davor Dragičević ◽  
Claudia Valls
Keyword(s):  
Lyapunov Exponents ◽  
Ergodic Theory ◽  
Lyapunov Theory ◽  
Type Result ◽  
Exponential Dichotomies ◽  
Linear Cocycle

We establish a Perron-type result for the perturbations of a linear cocycle in the context of ergodic theory. More precisely, we show that the Lyapunov exponents of a linear cocycle are preserved under sufficiently small nonautonomous perturbations. Our approach is based on the Lyapunov theory of regularity.

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