Nonuniform Exponential Dichotomy Research Articles

The relation between μ-dichotomy spectrum and Sacker-Sell spectrum

We present a relation between μ-dichotomy spectrum and Sacker-Sell spectrum. From this relation, we show that, under some suitable conditions, if the system x′=A(t)x admits a μ-dichotomy, then it admits an exponential dichotomy. Utilizing this result, we present a linearization theorem. Furthermore, we prove that the μ-dichotomy spectrum of the system x′=diag(A1(t),A2(t))x is the union of μ-dichotomy spectra of the subsystems x1′=A1(t)x1 and x2′=A2(t)x2, while this result is wrong if we replace μ-dichotomy spectrum with nonuniform exponential dichotomy spectrum. We also provide a new method to prove the μ-dichotomy spectral theorem.

Admissibility and nonuniform exponential dichotomies for difference equations without bounded growth or Lyapunov norms

Many previous works used the admissibility of function classes to characterize nonuniform exponential dichotomy (for short, NEDs) by employing Lyapunov norms. Recently, L. Zhou and W. Zhang [J. Funct. Anal. 271 (2016), pp. 1087–1129] characterized NEDs without Lyapunov norms. They utilized two admissible pairs of function classes and an assumption on certain subspaces to describe NEDs under the prerequisite of bounded growth, which plays an essential role in their arguments. However, in this paper, we remove the condition of bounded growth when characterizing NEDs. Neither bounded growth nor Lyapunov norms are used to describe NEDs in this paper.

Nonuniform Dichotomy with Growth Rates of Skew-Evolution Cocycles in Banach Spaces

This paper presents integral charaterizations for nonuniform dichotomy with growth rates and their correspondents for the particular cases of nonuniform exponential dichotomy and nonuniform polynomial dichotomy of skew-evolution cocycles in Banach spaces. The connections between these three concepts are presented.

Barbashin type characterizations for nonuniform <i>h</i>-dichotomy of evolution families

<abstract> <p>The aim of this paper is to give some Barbashin type characterizations for the nonuniform <italic>h</italic>-dichotomy of reversible evolution families in Banach spaces. Two necessary and sufficient conditions for the nonuniform <italic>h</italic>-dichotomy are pointed out using some important sets of growth functions. Additionally, as particular cases, we obtain a Barbashin type characterization for nonuniform exponential dichotomy and a necessary and sufficient condition for the nonuniform polynomial dichotomy.</p> </abstract>

Smooth dependence of nonautonomous linearization on parameters

Barreira and Valls (2010) proved in [4] the C1 dependence of C0 linearization on parameters for nonuniformly hyperbolic diffeomorphisms, whose nonlinear parts depend on the parameters but linear parts do not. In this paper, we further allow the linear parts to depend on the parameters, where only Hölder dependence is possible and a difficulty arises: the projections in the nonuniform exponential dichotomy do not have explicit expressions. We will overcome the difficulty by giving a new result on the Hölder dependence of nonuniform exponential dichotomy and constructing the conjugacy in a different way. Moreover, under Barreira and Valls's framework we will improve the C1 dependence to Cr, where another difficulty arises: without the non-resonant condition, the systems do not have small derivatives of orders ≥2 in general. We will introduce functional spaces with small bounds to overcome the difficulty. Besides, our different constructing method ensures that the conjugacy is invertible, which was not considered by Barreira and Valls.

Permanence of nonuniform nonautonomous hyperbolicity for infinite-dimensional differential equations

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.

Admissibility and nonuniform exponential dichotomies

Nonuniform exponential dichotomy represents the nonuniform hyperbolicity for evolution family. Considerable efforts have been made to the relationship between the existence of nonuniform exponential dichotomies and the admissibility of a pair of function classes. In order to reduce the effect of nonuniformity, most results on this aspect need Lyapunov norms in function classes. However, it is difficult to construct an appropriate Lyapunov norm before the existence of a nonuniform exponential dichotomy is established, which happens in the converse problem from admissibility to dichotomy. In this paper, we prove the relationship without a Lyapunov norm. Then we apply our results to giving the relationship between nonuniform exponential dichotomy and invertibility of a differential mapping on some function spaces.

Admissibility and generalized nonuniform dichotomies for discrete dynamics

<p style='text-indent:20px;'>We obtain characterizations of nonuniform dichotomies, defined by general growth rates, based on admissibility conditions. Additionally, we use the obtained characterizations to derive robustness results for the considered dichotomies. As particular cases, we recover several results in the literature concerning nonuniform exponential dichotomies and nonuniform polynomial dichotomies as well as new results for nonuniform dichotomies with logarithmic growth.</p>

Smooth Invariant Manifolds for Differential Equations with Infinite Delay

In this article, we give the existence of a smooth stable manifold which is invariant under the semiflows of the delay differential equation $ x^{\prime }= Ax(t) + Lx_t + f(t,x_t, \lambda ) $ , with the assumption that the corresponding linear differential equation admits a nonuniform exponential dichotomy and the perturbation f(t, xt, λ) is small and smooth enough. We also show that the obtained manifold is Lipschitz in the parameter λ.

Admissibility criteria for nonuniform dichotomic behavior of nonautonomous systems on the whole line

We give new criteria for nonuniform dichotomy of nonautonomous systems on the whole line in terms of admissibility relative to an integral equation. In our approach the input space I(R,X) is an intersection of spaces that can be successively minimized and the output space C(R,X) can be one of some well-known spaces of continuous functions. Using computational arguments, we show that the admissibility of (C(R,X),I(R,X)) leads to a nonuniform exponential dichotomy. We expose a complete analysis of the connections between admissibility and nonuniform dichotomy on the whole line and we also discuss several interesting consequences. Moreover, we obtain the explicit expression of the growth rates for dichotomy in terms of the initial exponential growth and the norm of the input-output operator. Finally, we present a direct application of the main result in the case of evolution families which admit uniform exponential growth.

Admissibility in the Strong and Weak Senses

For a dynamics with continuous time determined by a nonautonomous linear equation, we characterize completely the notion of an exponential dichotomy with respect to a family of norms in terms of an admissibility property. Moreover, we consider both strong and weak admissibility properties, which are expressed, respectively, in terms of classical and mild solutions. As a nontrivial application, we establish the robustness of the notions of an exponential dichotomy with respect to a family of norms and of a nonuniform exponential dichotomy in a very simple manner.

Nonuniform almost reducibility of nonautonomous linear differential equations

We prove that a linear nonautonomous differential system with nonuniform hyperbolicity on the half line can be written as diagonal system with a perturbation which is small enough. Moreover we show that the diagonal terms are contained in the nonuniform exponential dichotomy spectrum. For this purpose we introduce the concepts of nonuniform almost reducibility and nonuniform contractibility which are generalizations of these notions originally defined in a uniform context.

Linearization of a nonautonomous unbounded system with nonuniform contraction: A spectral approach

For a nonautonomous linear system with nonuniform contraction, we construct a topological conjugacy between this system and an unbounded nonlinear perturbation. This topological conjugacy is constructed as a composition of homeomorphisms. The first one is set up by considering the fact that linear system is almost reducible to diagonal system with a small enough perturbation where the diagonal entries belong to spectrum of the nonuniform exponential dichotomy; and the second one is constructed in terms of the crossing times with respect to unit sphere of an adequate Lyapunov function associated to the linear system.

Does nonuniform behavior destroy the structural stability?

This paper provides an answer if the nonuniform behavior can destroy the structural stability of nonlinear systems. We show that if the linear system $\dot{x}(t) = A(t)x(t)$ admits a nonuniform exponential dichotomy, then the perturbed nonautonomous system $\dot{x}(t) = A(t)x(t)+f(t, x)$ is structurally stable under suitable conditions.

Nonuniform exponential dichotomy for block triangular systems on the half line

Nonuniform exponential dichotomy for block triangular systems on the half line

NONUNIFORM DICHOTOMY SPECTRUM INTERVALS: THEOREM AND COMPUTATION

In this paper, on the condition of nonuniformly bounded growth, we give the relationship between the nonuniform exponential dichotomy spectrum and the other two classical spectrums (the Lyapunov spectrum and Sacker-Sell spectrum), present their stability under small linear perturbations. The main goal of this paper is to discuss the theory for the computation of these spectrums on the condition of nonuniformly bounded growth, and this extends the work of Dieci and Vleck [<span class="xref"><a href="#b22" ref-type="bibr">22</a></span>], which computes the Lyapunov spectrum and Sacker-Sell spectrum under the condition of bounded. Finally, the numerical examples are given to illustrate and verify the theoretical results.

Normal forms via nonuniform hyperbolicity

We develop a normal form theory for a nonautonomous dynamics xm+1=Amxm+fm(xm) with discrete time, based on the nonuniform spectrum of the sequence of matrices Am. In particular, we show that any nonresonant terms of the perturbations fm can be eliminated through an appropriate coordinate change, with the resonances expressed in terms of the connected components of the nonuniform spectrum. The latter is defined in terms of the notion of a nonuniform exponential dichotomy with a small nonuniform part, which is ubiquitous in the context of ergodic theory. We first make a preparation of the linear part of the dynamics that is of independent interest: we show that any sequence of matrices with a bounded nonuniform spectrum can be reduced to a sequence of matrices in block form via a Lyapunov coordinate change. This allows maintaining the Lyapunov exponents as well as the nonuniform spectrum. As further developments, we describe normal form theories in two additional contexts: we consider nonuniformly hyperbolic cocycles over a diffeomorphism of a compact manifold as well as perturbations fm of a sequence of compact linear operators Am on a Banach space. The latter includes the particular case of a sequence of matrices that need not be invertible.

A note on the nonuniform exponential stability and dichotomy for nonautonomous difference equations

In their recent result Hu and Mitsui proved that any exponentially stable nonautonomous linear difference equation can be reduced via suitable change of variables to equivalent linear difference equations whose coefficient matrices have spectral norm which is less then 1. In the present paper, we prove that this statement holds under weaker assumption that the nonautonomous linear difference equation is (strongly) nonuniformly exponentially stable. Moreover, we also establish corresponding results in the case of strong nonuniform exponential dichotomies.

Stable invariant manifolds for delay equations with piecewise constant argument

We construct smooth stable invariant manifolds for a class of delay equations with piecewise constant delay, for any sufficiently small perturbation of a nonuniform exponential dichotomy. We build on former work for perturbations of a uniform exponential dichotomy, also for delay equations with piecewise constant delay. These equations can be described as delay equations with an impulsive behavior of the derivative, such that at certain times the derivative changes abruptly.

Strong nonuniform behaviour: A Datko type characterization

We obtain a Datko-type characterization of several classes of a strong nonuniform exponential behaviour. We consider separately the case of exponential contractions, expansions and dichotomies. In addition, we consider both discrete and continuous time. As a nontrivial application of our work, we study the robustness property of strong nonuniform exponential contractions, expansions and dichotomies.