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Advanced Nonlinear Studies

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Volume 19, Issue 2


Shadowing for Nonautonomous Dynamics

Lucas Backes
  • Departamento de Matemática, Universidade Federal do Rio Grande do Sul, Av. Bento Gonçalves 9500, CEP 91509-900, Porto Alegre, RS, Brazil
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/ Davor Dragičević
Published Online: 2018-10-31 | DOI: https://doi.org/10.1515/ans-2018-2033


We prove that whenever a sequence of bounded operators (Am)m acting on a Banach space X admits an exponential dichotomy and a sequence of differentiable maps fm:XX, m, has bounded and Hölder derivatives, the nonautonomous dynamics given by xm+1=Amxm+fm(xm), m, 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.

Keywords: Shadowing; Nonautonomous Systems; Exponential Dichotomies; Nonlinear Perturbations

MSC 2010: 37C50; 34D09; 34D10


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About the article

Received: 2018-07-28

Revised: 2018-08-22

Accepted: 2018-09-17

Published Online: 2018-10-31

Published in Print: 2019-05-01

L. Backes was partially supported by a CAPES-Brazil postdoctoral fellowship under Grant No. 88881.120218/2016-01 at the University of Chicago. D. Dragičević was supported in part by the Croatian Science Foundation under the project IP-2014-09-2285 and by the University of Rijeka under the project number

Citation Information: Advanced Nonlinear Studies, Volume 19, Issue 2, Pages 425–436, ISSN (Online) 2169-0375, ISSN (Print) 1536-1365, DOI: https://doi.org/10.1515/ans-2018-2033.

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