Nonlinear perturbations of nonuniform exponential dichotomy on measure chains

2012 ◽  
Vol 75 (2) ◽  
pp. 670-683 ◽  
Author(s):  
Jimin Zhang ◽  
Meng Fan ◽  
Xiaoyuan Chang
Keyword(s):  
Exponential Dichotomy ◽  
Nonlinear Perturbations ◽  
Nonuniform Exponential Dichotomy

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).

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 Roughness of(ℤ+,ℤ-)-Nonuniform Exponential Dichotomy for Difference Equations in Banach Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Nicolae Lupa
Keyword(s):  
Banach Spaces ◽  
Explicit Form ◽  
Difference Equations ◽  
Exponential Dichotomy ◽  
General Context ◽  
Boundedness Condition ◽  
Infinite Dimensional ◽  
Perturbed Equation ◽  
Nonuniform Exponential Dichotomy

In this paper we study the roughness of(ℤ+,ℤ-)-nonuniform exponential dichotomy for nonautonomous difference equations in the general context of infinite-dimensional spaces. An explicit form is given for each of the dichotomy constants of the perturbed equation in terms of the original ones. We emphasize that we do not assume any boundedness condition on the coefficients.

Lyapunov theorems for exponential dichotomies in Hilbert spaces

2016 ◽  
Vol 27 (04) ◽  
pp. 1650033 ◽  
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.

scholarly journals Admissibility and nonuniform exponential dichotomy on the half-line

2013 ◽  
Vol 137 (4) ◽  
pp. 466-484 ◽  
Author(s):  
Adina Luminiţa Sasu ◽  
Mihai Gabriel Babuţia ◽  
Bogdan Sasu
Keyword(s):  
Exponential Dichotomy ◽  
Half Line ◽  
Nonuniform Exponential Dichotomy

scholarly journals Equivalences between nonuniform exponential dichotomy and admissibility

2017 ◽  
Vol 262 (1) ◽  
pp. 682-747 ◽  
Author(s):  
Linfeng Zhou ◽  
Kening Lu ◽  
Weinian Zhang
Keyword(s):  
Exponential Dichotomy ◽  
Nonuniform Exponential Dichotomy
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