Hostname: page-component-848d4c4894-m9kch Total loading time: 0 Render date: 2024-05-01T06:01:40.027Z Has data issue: false hasContentIssue false
  • English
  • Français

Multiscale linearization of nonautonomous systems

Part of: Stability theory

Published online by Cambridge University Press:  23 September 2022

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:
Faculty of Mathematics, University of Rijeka, Croatia (ddragicevic@math.uniri.hr)
Get access Rights & Permissions [Opens in a new window]

Abstract

We present sufficient conditions under which a given linear nonautonomous system and its nonlinear perturbation are topologically conjugated. Our conditions are of a very general form and provided that the nonlinear perturbations are well-behaved, we do not assume any asymptotic behaviour of the linear system. Moreover, the control on the nonlinear perturbations may differ along finitely many mutually complementary directions. We consider both the cases of one-sided discrete and continuous dynamics.

Keywords

Multiscalelinearizationnonautonomous dynamics

MSC classification

Primary: 34D30: Structural stability and analogous concepts
Secondary: 34D09: Dichotomy, trichotomy
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 153 , Issue 5 , October 2023 , pp. 1609 - 1629
Check for updates
Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aulbach, B. and Wanner, T.. Topological simplification of nonautonomous difference equations. J. Differ. Equ. Appl. 12 (2006), 283–296.10.1080/10236190500489384CrossRefGoogle Scholar
Backes, L. and Dragičević, D.. Quasi-shadowing for partially hyperbolic dynamics on Banach spaces. J. Math. Anal. Appl. 492 (2020), 124445.10.1016/j.jmaa.2020.124445CrossRefGoogle Scholar
Backes, L. and Dragičević, D.. A generalized Grobman-Hartman theorem for nonautonomous dynamics. Collect. Math. 73 (2022), 411–431.10.1007/s13348-021-00327-4CrossRefGoogle Scholar
Backes, L. and Dragičević, D., Smooth linearization of nonautonomous coupled systems, preprint arXiv:2202.12367.Google Scholar
Backes, L., Dragičević, D. and Palmer, K.. Linearization and Hölder continuity for nonautonomous systems. J. Differ. Equ. 297 (2021), 536–574.CrossRefGoogle Scholar
Castañeda, Á., González, P. and Robledo, G.. Topological equivalence of nonautonomous difference equations with a family of dichotomies on the half line. Comm. Pure Appl. Anal. 20 (2021), 511–532.10.3934/cpaa.2020278CrossRefGoogle Scholar
Castañeda, A. and Jara, N., A note on the differentiability of Palmer's topological equivalence for discrete systems, e-print https://arxiv.org/pdf/2104.14592.pdf.Google Scholar
Castaneda, A., Monzon, P. and Robledo., G. Nonuniform contractions and density stability results via a smooth topological equivalence, preprint https://arxiv.org/abs/1808.07568.Google Scholar
Castañeda, A. and Robledo, G.. Differentiability of Palmer's linearization theorem and converse result for density function. J. Differ. Equ. 259 (2015), 4634–4650.10.1016/j.jde.2015.06.004CrossRefGoogle Scholar
Castañeda, Á. and Robledo, G.. Dichotomy spectrum and almost topological conjugacy on nonautonomous unbounded difference systems. Discrete Contin. Dyn. Syst. 38 (2018), 2287–2304.10.3934/dcds.2018094CrossRefGoogle Scholar
Cuong, L. V., Doan, T. S. and Siegmund, S.. A Sternberg theorem for nonautonomous differential equations. J. Dynam. Diff. Eq. 31 (2019), 1279–1299.10.1007/s10884-017-9629-8CrossRefGoogle Scholar
Coppel, W. A.. Dichotomies in Stability Theory (Berlin, Heidelberg, New-York: Springer Verlag, 1978).10.1007/BFb0067780CrossRefGoogle Scholar
Dragičević, D.. Global smooth linearization of nonautonomous contractions on Banach spaces. Electron. J. Qual. Theory Differ. Equ. 12 (2022), 1–19.10.14232/ejqtde.2022.1.12CrossRefGoogle Scholar
Dragičević, D., Zhang, W. and Zhang, W.. Smooth linearization of nonautonomous difference equations with a nonuniform dichotomy. Math. Z. 292 (2019), 1175–1193.10.1007/s00209-018-2134-xCrossRefGoogle Scholar
Dragičević, D., Zhang, W. and Zhang, W.. Smooth linearization of nonautonomous differential equations with a nonuniform dichotomy. Proc. Lond. Math. Soc. 121 (2020), 32–50.10.1112/plms.12315CrossRefGoogle Scholar
Grobman, D.. Homeomorphism of systems of differential equations. Dokl. Akad. Nauk SSSR 128 (1959), 880–881.Google Scholar
Grobman, D.. Topological classification of neighborhoods of a singularity in $n$-space. Mat. Sb. (N.S.) 56 (1962), 77–94.Google Scholar
Hartman, P.. On local homeomorphisms of Euclidean spaces. Bol. Soc. Mat. Mexicana (2), 5 (1960), 220–241.Google Scholar
Hartman, P.. A lemma in the theory of structural stability of differential equations. Proc. Amer. Math. Soc. 11 (1960), 610–620.10.1090/S0002-9939-1960-0121542-7CrossRefGoogle Scholar
Hartman, P.. On the local linearization of differential equations. Proc. Amer. Math. Soc. 14 (1963), 568–573.10.1090/S0002-9939-1963-0152718-3CrossRefGoogle Scholar
Jiang, L.. Generalized exponential dichotomy and global linearization. J. Math. Anal. Appl. 315 (2006), 474–490.10.1016/j.jmaa.2005.05.042CrossRefGoogle Scholar
Lin, F.. Hartman's linearization on nonautonomous unbounded system. Nonlinear Anal. 66 (2007), 38–50.10.1016/j.na.2005.11.007CrossRefGoogle Scholar
Palis, J.. On the local structure of hyperbolic points in Banach spaces. An. Acad. Brasil. Cienc. 40 (1968), 263–266.Google Scholar
Palmer, K.. A generalization of Hartman's linearization theorem. J. Math. Anal. Appl. 41 (1973), 753–758.CrossRefGoogle Scholar
Pilyugin, S. Y.. Multiscale conditional shadowing. J. Dyn. Diff. Eq. (2021). doi:10.1007/s10884-021-10096-0CrossRefGoogle Scholar
Pugh, C.. On a theorem of P. Hartman. Amer. J. Math. 91 (1969), 363–367.10.2307/2373513CrossRefGoogle Scholar
Reinfelds, A. and Šteinberga, D.. Dynamical equivalence of quasilinear equations. Intern. J. Pure and Appl. Math. 98 (2015), 355–364.CrossRefGoogle Scholar
Sternberg, S.. Local contractions and a theorem of Poincaré. Amer. J. Math. 79 (1957), 809–824.10.2307/2372437CrossRefGoogle Scholar
Sternberg, S.. On the structure of local homeomorphisms of Euclidean $n$-space. Amer. J. Math. 80 (1958), 623–631.10.2307/2372774CrossRefGoogle Scholar