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Shadowing for infinite dimensional dynamics and exponential trichotomies

Part of: Stability theory Smooth dynamical systems: general theory

Published online by Cambridge University Press:  24 June 2020

Lucas Backes  and
Davor Dragičević
Lucas Backes
Affiliation:
Departamento de Matemática, Universidade Federal do Rio Grande do Sul, Av. Bento Gonçalves 9500, CEP 91509-900 Porto Alegre, RS, Brazil (lucas.backes@ufrgs.br)
Davor Dragičević
Affiliation:
Department of Mathematics, University of Rijeka, Croatia (ddragicevic@math.uniri.hr)
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Abstract

Let $(A_m)_{m \in {\mathop Z}}$ be a sequence of bounded linear maps acting on an arbitrary Banach space X and admitting an exponential trichotomy and let $f_m:X \to X$ be a Lispchitz map for every $m\in {\mathop Z} $. We prove that whenever the Lipschitz constants of $f_m$, $m \in {\mathop Z} $, are uniformly small, the nonautonomous dynamics given by $x_{m+1}=A_mx_m+f_m(x_m)$, $m\in {\mathop Z} $, has various types of shadowing. Moreover, if X is finite dimensional and each $A_m$ is invertible we prove that a converse result is also true. Furthermore, we get similar results for one-sided and continuous time dynamics. As applications of our results, we study the Hyers–Ulam stability for certain difference equations and we obtain a very general version of the Grobman–Hartman's theorem for nonautonomous dynamics.

Keywords

ShadowingNonautonomus systemsExponential trichotomiesNonlinear perturbationsHyers–Ulam stability

MSC classification

Primary: 37C50: Approximate trajectories (pseudotrajectories, shadowing, etc.) 34D09: Dichotomy, trichotomy
Secondary: 34D10: Perturbations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 3 , June 2021 , pp. 863 - 884
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Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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