Exponential dichotomies with respect to a sequence of norms and admissibility

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.

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Lyapunov theorems for exponential dichotomies in Hilbert spaces

International Journal of Mathematics ◽

10.1142/s0129167x16500336 ◽

2016 ◽

Vol 27 (04) ◽

pp. 1650033 ◽

Cited By ~ 3

Author(s):

Davor Dragičević ◽

Ciprian Preda

Keyword(s):

Hilbert Spaces ◽

Exponential Dichotomy ◽

Complete Characterization ◽

Linear Operators ◽

Nonlinear Perturbations ◽

Exponential Behavior ◽

Exponential Dichotomies ◽

Linear And Nonlinear ◽

Unstable Direction

For a nonautonomous dynamics defined by a sequence of linear operators, we obtain a complete characterization of the notion of a uniform exponential dichotomy in terms of the existence of appropriate Lyapunov sequences. In sharp contrast to previous results, we consider the case of noninvertible dynamics, thus requiring only the invertibility of operators along the unstable direction. Furthermore, we deal with operators acting on an arbitrary Hilbert space. As a nontrivial application of our work, we study the persistence of uniform exponential behavior under small linear and nonlinear perturbations.

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NONUNIFORM EXPONENTIAL BEHAVIOUR AND TOPOLOGICAL EQUIVALENCE

Glasgow Mathematical Journal ◽

10.1017/s0017089515000191 ◽

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.

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Stable manifolds for perturbations of exponential dichotomies in mean

Stochastics and Dynamics ◽

10.1142/s0219493718500223 ◽

2018 ◽

Vol 18 (03) ◽

pp. 1850022 ◽

Cited By ~ 1

Author(s):

Luis Barreira ◽

Claudia Valls

Keyword(s):

Nonlinear Dynamics ◽

Banach Space ◽

Exponential Dichotomy ◽

Invariant Manifolds ◽

Upper Bounds ◽

Lower And Upper Bounds ◽

Exponential Behavior ◽

Exponential Dichotomies ◽

Stable Invariant Manifolds ◽

Stable Invariant

We establish the existence of stable invariant manifolds for any sufficiently small perturbation of a cocycle with an exponential dichotomy in mean. The latter notion corresponds to replace the exponential behavior in the classical notion of an exponential dichotomy by an exponential behavior in average with respect to an invariant measure. We consider both perturbations of a cocycle over a map and over a flow that can be defined on an arbitrary Banach space. Moreover, we obtain an upper bound for the speed of the nonlinear dynamics along the stable manifold as well as a lower bound when the exponential dichotomy in mean is strong (this means that we have lower and upper bounds along the stable and unstable directions of the dichotomy).

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Admissibility for exponential dichotomies in average

Stochastics and Dynamics ◽

10.1142/s0219493715500148 ◽

2015 ◽

Vol 15 (03) ◽

pp. 1550014 ◽

Cited By ~ 1

Author(s):

Luis Barreira ◽

Davor Dragičević ◽

Claudia Valls

Keyword(s):

Probability Measure ◽

Linear Operator ◽

Exponential Dichotomy ◽

Simple Manner ◽

Exponential Dichotomies ◽

Linear Perturbations ◽

Bounded Sequences

We characterize completely the notion of an exponential dichotomy in average in terms of an admissibility property. The notion corresponds to a generalization of that of an exponential dichotomy to measurable cocycles acting on L1 functions with respect to a given probability measure. The admissibility property is described in terms of the injectivity and surjectivity of a certain linear operator in the space of bounded sequences of L1 functions. The characterization is then used to establish in a simple manner the robustness of the notion, in the sense that it persists under sufficiently small linear perturbations. We note that we consider both ℤ-cocycles and ℕ-cocycles.

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Exponential behavior in mean for cocycles

Stochastics and Dynamics ◽

10.1142/s021949371550029x ◽

2015 ◽

Vol 15 (04) ◽

pp. 1550029

Author(s):

Luis Barreira ◽

Claudia Valls

Keyword(s):

Probability Measure ◽

Discrete Time ◽

Exponential Dichotomy ◽

Exponential Behavior ◽

Classical Notion ◽

Linear Perturbations

For cocycles with discrete time, we consider the notion of an exponential dichotomy in mean. This corresponds to replace the classical notion of an exponential dichotomy by the much weaker requirement that the same happens in mean with respect to some probability measure. We show that the exponential behavior in mean is robust, in the sense that it persists under sufficiently small linear perturbations.

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Strong nonuniform spectrum for arbitrary growth rates

Communications in Contemporary Mathematics ◽

10.1142/s0219199716500085 ◽

2017 ◽

Vol 19 (02) ◽

pp. 1650008 ◽

Cited By ~ 3

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.

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Permanence of nonuniform nonautonomous hyperbolicity for infinite-dimensional differential equations

Asymptotic Analysis ◽

10.3233/asy-211719 ◽

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.

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Smooth Stable Manifold for Delay Differential Equations with Arbitrary Growth Rate

Axioms ◽

10.3390/axioms10020105 ◽

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).

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Conjugacies between linear and nonlinear non-uniform contractions

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385707000636 ◽

2008 ◽

Vol 28 (1) ◽

pp. 1-19 ◽

Cited By ~ 3

Author(s):

LUIS BARREIRA ◽

CLAUDIA VALLS

Keyword(s):

Banach Spaces ◽

Discrete Time ◽

Linear Operators ◽

Nonlinear Perturbations ◽

Exponential Dichotomies ◽

Locally Lipschitz ◽

Linear And Nonlinear

AbstractWe construct conjugacies between linear and nonlinear non-uniform exponential contractions with discrete time. We also consider the general case of a non-autonomous dynamics defined by a sequence of maps. The results are obtained by considering both linear and nonlinear perturbations of the dynamics xm+1=Amxm defined by a sequence of linear operators Am. In the case of conjugacies between linear contractions we describe them explicitly. All the conjugacies are locally Hölder, and in fact are locally Lipschitz outside the origin. We also construct conjugacies between linear and nonlinear non-uniform exponential dichotomies, building on the arguments for contractions. All the results are obtained in Banach spaces.

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Nonuniform Hyperbolicity and Admissibility

Advanced Nonlinear Studies ◽

10.1515/ans-2014-0315 ◽

2014 ◽

Vol 14 (3) ◽

Cited By ~ 12

Author(s):

Luis Barreira ◽

Claudia Valls ◽

Davor Dragičevič

Keyword(s):

Exponential Dichotomy ◽

Linear Operators ◽

Nonuniform Hyperbolicity

AbstractFor a nonautonomous dynamics defined by a sequence of linear operators, we consider the notion of an exponential dichotomy with respect to a sequence of norms and we characterize it completely in terms of the admissibility in pairs of spaces (ℓ

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