Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Advanced Nonlinear Studies

Editor-in-Chief: Ahmad, Shair


IMPACT FACTOR 2017: 1.029
5-year IMPACT FACTOR: 1.147

CiteScore 2017: 1.29

SCImago Journal Rank (SJR) 2017: 1.588
Source Normalized Impact per Paper (SNIP) 2017: 0.971

Mathematical Citation Quotient (MCQ) 2017: 1.03

Online
ISSN
2169-0375
See all formats and pricing
More options …
Volume 19, Issue 2

Issues

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
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Davor Dragičević
Published Online: 2018-10-31 | DOI: https://doi.org/10.1515/ans-2018-2033

Abstract

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

References

  • [1]

    D. Anosov, On a certain class of invariant sets of smooth dynamical systems (in Russian), Analytical Methods of the Theory of Nonlinear Oscillations (Kiev 1969), Akademie der Wissenschaften der Ukrainischen SSR, Kiev (1970), 39–45. Google Scholar

  • [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. Google Scholar

  • [3]

    L. Barreira, D. Dragičević and C. Valls, Nonuniform spectrum on Banach spaces, Adv. Math. 321 (2017), 547–591. CrossrefWeb of ScienceGoogle Scholar

  • [4]

    L. Barreira and C. Valls, A Grobman–Hartman theorem for nonuniformly hyperbolic dynamics, J. Differential Equations 228 (2006), no. 1, 285–310. CrossrefGoogle Scholar

  • [5]

    L. Barreira and C. Valls, Hölder Grobman–Hartman linearization, Discrete Contin. Dyn. Syst. 18 (2007), no. 1, 187–197. CrossrefGoogle Scholar

  • [6]

    N. C. Bernardes, Jr., P. R. Cirilo, U. B. Darji, A. Messaoudi and E. R. Pujals, Expansivity and shadowing in linear dynamics, J. Math. Anal. Appl. 461 (2018), no. 1, 796–816. CrossrefWeb of ScienceGoogle Scholar

  • [7]

    C. Bonatti, L. J. Díaz and G. Turcat, Pas de “shadowing lemma” pour des dynamiques partiellement hyperboliques, C. R. Acad. Sci. Paris Sér. I Math. 330 (2000), no. 7, 587–592. CrossrefGoogle Scholar

  • [8]

    R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Math. 470, Springer, Berlin, 1975. Google Scholar

  • [9]

    S.-N. Chow, X.-B. Lin and K. J. Palmer, A shadowing lemma with applications to semilinear parabolic equations, SIAM J. Math. Anal. 20 (1989), no. 3, 547–557. CrossrefGoogle Scholar

  • [10]

    D. Dragičević, Admissibility, a general type of Lipschitz shadowing and structural stability, Commun. Pure Appl. Anal. 14 (2015), no. 3, 861–880. Web of ScienceCrossrefGoogle Scholar

  • [11]

    D. Dragičević, W. Zhang and W. Zhang, Smooth linearization of nonautonomous difference equations with a nonuniform dichotomy, Math. Z. (2018), 10.1007/s00209-018-2134-x. Google Scholar

  • [12]

    S. Gan, A generalized shadowing lemma, Discrete Contin. Dyn. Syst. 8 (2002), no. 3, 627–632. CrossrefGoogle Scholar

  • [13]

    P. Hartman, On local homeomorphisms of Euclidean spaces, Bol. Soc. Mat. Mex. (2) 5 (1960), 220–241. Google Scholar

  • [14]

    D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math. 840, Springer, Berlin, 1981. Google Scholar

  • [15]

    D. B. Henry, Exponential dichotomies, the shadowing lemma and homoclinic orbits in Banach spaces, Resenhas 1 (1994), no. 4, 381–401. Google Scholar

  • [16]

    A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Publ. Math. Inst. Hautes Études Sci. (1980), no. 51, 137–173. Google Scholar

  • [17]

    A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia Math. Appl. 54, Cambridge University Press, Cambridge, 1995. Google Scholar

  • [18]

    H. Koçak, K. Palmer and B. Coomes, Shadowing in ordinary differential equations, Rend. Semin. Mat. Univ. Politec. Torino 65 (2007), no. 1, 89–113. Google Scholar

  • [19]

    O. E. Lanford, Introduction to hyperbolic sets, Regular and Chaotic Motions in Dynamical Systems, Springer, Boston (1985), 73–102. Google Scholar

  • [20]

    K. R. Meyer and G. R. Sell, An analytic proof of the shadowing lemma, Funkcial. Ekvac. 30 (1987), no. 1, 127–133. Google Scholar

  • [21]

    K. Palmer, Shadowing in Dynamical Systems. Theory and Applications, Math. Appl. 501, Kluwer Academic, Dordrecht, 2000. Google Scholar

  • [22]

    K. J. Palmer, A generalization of Hartman’s linearization theorem, J. Math. Anal. Appl. 41 (1973), 753–758. CrossrefGoogle Scholar

  • [23]

    K. J. Palmer, Exponential dichotomies, the shadowing lemma and transversal homoclinic points, Dynamics Reported. Vol. 1, Dynam. Report. Ser. Dynam. Systems Appl. 1, Wiley, Chichester, (1988), 265–306. Google Scholar

  • [24]

    O. Perron, Die Stabilitätsfrage bei Differentialgleichungen, Math. Z. 32 (1930), no. 1, 703–728. CrossrefGoogle Scholar

  • [25]

    S. Y. Pilyugin, Shadowing in structurally stable flows, J. Differential Equations 140 (1997), no. 2, 238–265. CrossrefGoogle Scholar

  • [26]

    S. Y. Pilyugin, Shadowing in Dynamical Systems, Lecture Notes in Math. 1706, Springer, Berlin, 1999. Google Scholar

  • [27]

    A. L. Sasu, Exponential dichotomy and dichotomy radius for difference equations, J. Math. Anal. Appl. 344 (2008), no. 2, 906–920. CrossrefWeb of ScienceGoogle Scholar

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 17.15.2.2.01.


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.

Export Citation

© 2019 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in