Exponential dichotomies

2012 ◽  
pp. 187-222
Author(s):  
Alexandre N. Carvalho ◽  
José A. Langa ◽  
James C. Robinson
Keyword(s):  
Exponential Dichotomies

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 Fredholm properties of Lu = u′ − A(t)u for paths of sectorial operators

2005 ◽  
Vol 135 (1) ◽  
pp. 39-60 ◽  
Author(s):  
Davide di Giorgio ◽  
Alessandra Lunardi
Keyword(s):  
Banach Space ◽  
Explicit Formula ◽  
Fredholm Operator ◽  
Finite Dimension ◽  
Sufficient Conditions ◽  
Spectral Flow ◽  
Exponential Dichotomies ◽  
Sectorial Operators ◽  
Common Domain ◽  
General Banach Space

We consider a path of sectorial operators t ↦ A (t) ∈ Cα (R, L (D, X)), 0 < α < 1, in general Banach space X, with common domain D (A (t)) = D and with hyperbolic limits at ±∞. We prove that there exist exponential dichotomies in the half-lines (−∞, −T] and [T, +∞) for large T, and we study the operator (Lu)(t) = u′(t) − A(t)u(t) in the space Cα (R, D) ∩ C1+α (R, X). In particular, we give sufficient conditions in order that L is a Fredholm operator. In this case, the index of L is given by an explicit formula, which coincides to the well-known spectral flow formula in finite dimension. Such sufficient conditions are satisfied, for instance, if the embedding D ↪ X is compact.

Tempered exponential dichotomies: admissibility and stability under perturbations

2016 ◽  
Vol 31 (4) ◽  
pp. 525-545 ◽  
Author(s):  
Luis Barreira ◽  
Davor Dragičević ◽  
Claudia Valls
Keyword(s):  
Exponential Dichotomies ◽  
Stability Under Perturbations

Conjugacies between linear and nonlinear non-uniform contractions

2008 ◽  
Vol 28 (1) ◽  
pp. 1-19 ◽  
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.

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.

Exponential Dichotomies: Continuous Time

2018 ◽  
pp. 55-74
Author(s):  
Luís Barreira ◽  
Davor Dragičević ◽  
Claudia Valls
Keyword(s):  
Continuous Time ◽  
Exponential Dichotomies

Exponential Dichotomies: Discrete Time

2018 ◽  
pp. 33-54
Author(s):  
Luís Barreira ◽  
Davor Dragičević ◽  
Claudia Valls
Keyword(s):  
Discrete Time ◽  
Exponential Dichotomies
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