Almost reducibility of linear difference systems from a spectral point of view

Castañeda, Álvaro; Robledo, Gonzalo

doi:10.3934/cpaa.2017097

Cited by 4 publications

(26 citation statements)

References 14 publications

(12 reference statements)

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“…In [17] it is introduced the concept of diagonal significance which is fundamental for obtain the almost reducibility in the the case of exponential dichotomy [7,Proposition 4] in a discrete context.…”

Section: Preparatory Resultsmentioning

confidence: 99%

“…A discrete version of this notion was given byÁ. Castañeda and G. Robledo (see [7]). Now we introduce the definition of nonuniformly almost reducible, which is a version of the previous concept in the nonuniform framework.…”

Section: Preliminariesmentioning

confidence: 99%

“…As we said previously, the concept of almost reducibility was introduced in the 60's by B. F. Bylov in the continuous context. A discrete version of this notion was given by Á. Castañeda and G. Robledo (see [7]). Now we introduce the definition of nonuniformly almost reducible, which is a version of the previous concept in the nonuniform framework.…”

Section: Definition 1 ([22])mentioning

confidence: 99%

“…The concept of almost reducibility to diagonal system was rediscovered and improved in the 90's by F. Lin in [14], who introduces the concept of contractibility in the continuous context, while in the discrete case was proposed by Á. Castañeda and G. Robledo in [7]. In this paper we introduce its nonuniform version.…”

Section: Definition 1 ([22])mentioning

confidence: 99%

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Nonuniform almost reducibility of nonautonomous linear differential equations

Castañeda

Huerta

2020

Journal of Mathematical Analysis and Applications

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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 generalization of these notions originally defined in a uniform context.

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Section: Preparatory Resultsmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Section: Definition 1 ([22])mentioning

confidence: 99%

Section: Definition 1 ([22])mentioning

confidence: 99%

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Nonuniform almost reducibility of nonautonomous linear differential equations

Castañeda

Huerta

2020

Journal of Mathematical Analysis and Applications

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“…where C(n) is a diagonal matrix and B(n) is an upper triangular matrix satisfying ||B(n)|| < δ, where δ can be chosen arbitrarily small. This fact is motivated by a recent result of almost reducibility [6], which states that (1) can be transformed into (4) by a linear change of coordinates. The concept of reducibility is well known in the continuous case and we refer the reader to [8,9] for details.…”

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confidence: 99%

Dichotomy spectrum and almost topological conjugacy on nonautonomus unbounded difference systems

Castañeda¹,

Robledo²

2018

Discrete &Amp; Continuous Dynamical Systems - A

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We construct a bijection between the solutions of a linear system of nonautonomous difference equations which is uniformly asymptotically stable and its unbounded perturbation. The key idea used to made this bijection is to consider the crossing times of the solutions with the unit sphere. This approach prompt us to introduce the concept of almost topological conjugacy in this nonautonomous framework. This task is carried out by simplifying both systems through a spectral approach of the notion of almost reducibility combined with suitable technical assumptions.

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Is the Sacker–Sell type spectrum equal to the contractible set?

Wu¹,

Xia²

2023

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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For linear differential systems, the Sacker–Sell spectrum (dichotomy spectrum) and the contractible set are the same. However, we claim that this is not true for the linear difference equations. A counterexample is given. For the convenience of research, we study the relations between the dichotomy spectrum and the contractible set under the framework on time scales. In fact, by a counterexample, we show that the contractible set could be different from dichotomy spectrum on time scales established by Siegmund [J. Comput. Appl. Math., 2002]. Furthermore, we find that there is no bijection between them. In particular, for the linear difference equations, the contractible set is not equal to the dichotomy spectrum. To counter this mismatch, we propose a new notion called generalized contractible set and we prove that the generalized contractible set is exactly the dichotomy spectrum. Our approach is based on roughness theory and Perron's transformation. In this paper, a new method for roughness theory on time scales is provided. Moreover, we provide a time-scaled version of the Perron's transformation. However, the standard argument is invalid for Perron's transformation. Thus, some novel techniques should be employed to deal with this problem. Finally, an example is given to verify the theoretical results.

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Almost reducibility of linear difference systems from a spectral point of view

Cited by 4 publications

References 14 publications

Nonuniform almost reducibility of nonautonomous linear differential equations

Nonuniform almost reducibility of nonautonomous linear differential equations

Dichotomy spectrum and almost topological conjugacy on nonautonomus unbounded difference systems

Is the Sacker–Sell type spectrum equal to the contractible set?

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