Admissibility for exponential dichotomies in average

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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Exponential dichotomies with respect to a sequence of norms and admissibility

International Journal of Mathematics ◽

10.1142/s0129167x14500244 ◽

2014 ◽

Vol 25 (03) ◽

pp. 1450024 ◽

Cited By ~ 2

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.

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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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Conjugacies for impulsive equations

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091510000799 ◽

2011 ◽

Vol 55 (1) ◽

pp. 65-78

Author(s):

Luis Barreira ◽

Claudia Valls

Keyword(s):

Differential Equations ◽

Direct Proof ◽

Impulsive Differential Equations ◽

Nonlinear Perturbations ◽

Nonlinear Case ◽

Exponential Dichotomies ◽

Impulsive Equations ◽

Linear Perturbations ◽

Linear And Nonlinear ◽

Topological Conjugacies

AbstractFor impulsive differential equations, we construct topological conjugacies between linear and nonlinear perturbations of non-uniform exponential dichotomies. In the case of linear perturbations, the topological conjugacies are constructed in a more or less explicit manner. In the nonlinear case, we obtain an appropriate version of the Grobman–Hartman Theorem for impulsive equations, with a simple and direct proof that involves no discretization of the dynamics.

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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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On the Exponents of Exponential Dichotomies

Mathematics ◽

10.3390/math8040651 ◽

2020 ◽

Vol 8 (4) ◽

pp. 651

Author(s):

Flaviano Battelli ◽

Michal Fečkan

Keyword(s):

Differential Equations ◽

Exponential Dichotomy ◽

Nonlinear Differential Equations ◽

Linear Differential Equations ◽

Constructive Method ◽

Bifurcation Method ◽

Exponential Dichotomies ◽

The Difference ◽

Linear Differential

An exponential dichotomy is studied for linear differential equations. A constructive method is presented to derive a roughness result for perturbations giving exponents of the dichotomy as well as an estimate of the norm of the difference between the corresponding two dichotomy projections. This roughness result is crucial in developing a Melnikov bifurcation method for either discontinuous or implicit perturbed nonlinear differential equations.

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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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Almost Periodic Solutions for a Class of Stochastic Differential Equations

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4023914 ◽

2013 ◽

Vol 8 (4) ◽

Cited By ~ 3

Author(s):

Yongjian Liu ◽

Aimin Liu

Keyword(s):

Linear Operator ◽

Periodic Solutions ◽

Existence And Uniqueness ◽

Exponential Dichotomy ◽

Stochastic Equations ◽

Almost Periodic Solution ◽

Almost Periodic Solutions ◽

Almost Periodic ◽

Uniform Exponential Stability ◽

New Criteria

In this paper, the existence and uniqueness of the square-mean almost periodic solutions to a class of the semilinear stochastic equations is studied. In particular, the condition of the uniform exponential stability of the linear operator is essentially removed, only using the exponential dichotomy of the linear operator. Some new criteria ensuring the existence and uniqueness of the square-mean almost periodic solution for the system are presented. Finally, an example of a kind of the stochastic cellular neural networks is given. These obtained results are important in signal processing and the in design of networks.

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A perturbation theorem for exponential dichotomies

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500018175 ◽

1987 ◽

Vol 106 (1-2) ◽

pp. 25-37 ◽

Cited By ~ 19

Author(s):

Kenneth J. Palmer

Keyword(s):

Linear Equation ◽

Exponential Dichotomy ◽

Perturbation Theorem ◽

Exponential Dichotomies

SynopsisSuppose the linear equation x' = A(t)x has an exponential dichotomy and suppose B(t) is close to A(t) in the following sense: on any interval of length 2T, B(t) is close to some translate A(t + τ) of A(t) (actually the conditions in the paper are slightly weaker than this). Then if T is sufficiently large, the equation y' = B(t)y also has an exponential dichotomy. This generalises the usual roughness theorem for exponential dichotomies.

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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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