Shadowing for Nonautonomous Dynamics

Author:

Backes Lucas1,Dragičević Davor2

Affiliation:

1. Departamento de Matemática , Universidade Federal do Rio Grande do Sul , Av. Bento Gonçalves 9500, CEP 91509-900 , Porto Alegre , RS , Brazil

2. Department of Mathematics , University of Rijeka , Rijeka , Croatia

Abstract

Abstract We prove that whenever a sequence of bounded operators ( A m ) m {(A_{m})_{m\in\mathbb{Z}}} acting on a Banach space X admits an exponential dichotomy and a sequence of differentiable maps f m : X X {f_{m}\colon X\to X} , m {m\in\mathbb{Z}} , has bounded and Hölder derivatives, the nonautonomous dynamics given by x m + 1 = A m x m + f m ( x m ) {x_{m+1}=A_{m}x_{m}+f_{m}(x_{m})} , m {m\in\mathbb{Z}} , has various shadowing properties. Hence, we extend recent results of Bernardes Jr. et al. in several directions. As a nontrivial application of our results, we give a new proof of the nonautonomous Grobman–Hartman theorem.

Publisher

Walter de Gruyter GmbH

Subject

General Mathematics,Statistical and Nonlinear Physics

Reference27 articles.

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2. L. Barreira, D. Dragičević and C. Valls, Existence of conjugacies and stable manifolds via suspensions, Electron. J. Differential Equations 2017 (2017), Paper No. 172.

3. L. Barreira, D. Dragičević and C. Valls, Nonuniform spectrum on Banach spaces, Adv. Math. 321 (2017), 547–591. 10.1016/j.aim.2017.10.006

4. L. Barreira and C. Valls, A Grobman–Hartman theorem for nonuniformly hyperbolic dynamics, J. Differential Equations 228 (2006), no. 1, 285–310. 10.1016/j.jde.2006.04.001

5. L. Barreira and C. Valls, Hölder Grobman–Hartman linearization, Discrete Contin. Dyn. Syst. 18 (2007), no. 1, 187–197. 10.3934/dcds.2007.18.187

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