Hostname: page-component-54dcc4c588-sdd8f Total loading time: 0 Render date: 2025-10-04T01:23:14.068Z Has data issue: false hasContentIssue false
  • English
  • Français

Weighted Lipschitz shadowing and generalized weighted limit shadowing for infinite-dimensional systems

Part of: Stability theory Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  30 September 2025

Haiyang Huang  and
Linfeng Zhou
Haiyang Huang
Affiliation:
School of Mathematics, Sichuan University, Chengdu, Sichuan 610064, P.R. China (huanghy508@163.com)
Linfeng Zhou*
Affiliation:
School of Mathematics, Sichuan University, Chengdu, Sichuan 610064, P.R. China (zlf63math@163.com)
*
*Corresponding author.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we show that for any nonautonomous discrete time dynamical system in a Banach space if its linear part has a dichotomy and the composition of a generalized Green function and the nonlinear term of the system has a weighted integrable Lipschitz constant then the system has the weighted Lipschitz shadowing property for a type of weighted pseudo orbits in the whole phase space. Additionally, if the generalized Green function is the Green function for the dichotomy and the evolution operator restricted to the stable subspace (resp. unstable subspace) tends to 0 in weight as time tends to $+\infty$ (resp. $-\infty$) then the system has the weighted generalized forward (resp. backward) limit shadowing property. By the same approach we prove that a C1 map with a compact hyperbolic invariant set has the weighted Lipschitz shadowing property and the generalized weighted limit shadowing property for weighted pseudo orbits in the hyperbolic set. We also give the parallel results for differential equations.

Keywords

dichotomyinfinite dimensionlimit shadowinglipschitz shadowingpseudo orbit

MSC classification

Primary: 34D09: Dichotomy, trichotomy 37D05: Hyperbolic orbits and sets

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 39
Check for updates
Copyright
© The Author(s), 2025. 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.)

Article purchase

Temporarily unavailable

References

Akin, E.. The General Topology of Dynamical Systems. Graduate Studies in Mathematics, Vol. 1 (Providence, RI, 1993).Google Scholar
Anosov, D. V.. Geodesic flows on closed Riemannian manifolds of negative curvature. Trudy Mat. Inst. Steklov. 90 (1967), 3–210.Google Scholar
Backes, L. and Dragičević, D.. Shadowing for infinite dimensional dynamics and exponential trichotomies. Proc. Roy. Soc. Edinburgh Sect. A. 151 (2021), 863–884.CrossRefGoogle Scholar
Backes, L., Dragičević, D., Pituk, M. and Singh, L.. Weighted shadowing for delay differential equations. Arch. Math. 119 (2022), 539–552.CrossRefGoogle Scholar
Backes, L., Dragičević, D. and Singh, L.. Shadowing for nonautonomous and nonlinear dynamics with impulses. Monat. Math. 198 (2022), 485–502.10.1007/s00605-021-01629-2CrossRefGoogle Scholar
Bento, A. and Silva, C.. Nonuniform dichotomic behavior: Lipschitz invariant manifolds for ODEs. Bull. Sci. Math. 138 (2014), 89–109.10.1016/j.bulsci.2013.09.008CrossRefGoogle Scholar
Bowen, R.. ω-limit sets for axiom $\rm{A}$ diffeomorphisms. J. Differential Equ. 18 (1975), 333–339.10.1016/0022-0396(75)90065-0CrossRefGoogle Scholar
Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Lecture Notes in Mathematics, Vol. 470 (Springer, Berlin, 1975).10.1007/BFb0081279CrossRefGoogle Scholar
Chow, S. N., Lin, X. B. and Palmer, K. J.. A shadowing lemma with applications to semilinear parabolic equations. SIAM J. Math. Anal. 20 (1989), 547–557.10.1137/0520038CrossRefGoogle Scholar
Coppel, W. A.. Stability and Asymptotic Behavior of Differential Equations. (D.C. Heath and Company, Boston, 1965).Google Scholar
Daleckij, J. and Krein, M.. Stability of Solutions of Differential Equations in Banach Space. Translations of Mathematical Monographs, Vol. 43 (Providence, RI, 1974).Google Scholar
Elaydi, S. and Hajek, O.. Exponential trichotomy of differential systems. J. Math. Anal. Appl. 129 (1988), 362–374.10.1016/0022-247X(88)90255-7CrossRefGoogle Scholar
Hale, J. K.. Ordinary Diffential Equations. Krieger Publishing Company, Malabar, 1980.Google Scholar
Henry, D.. Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics, Vol. 840 (Springer, Berlin, 1981).10.1007/BFb0089647CrossRefGoogle Scholar
Henry, D.. Exponential dichotomies, the shadowing lemma and homoclinic orbits in Banach spaces. Resenhas. 1 (1994), 381–401.Google Scholar
Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems (Cambridge University Press, Cambridge, 1995).10.1017/CBO9780511809187CrossRefGoogle Scholar
Katok, A.. Nonuniform hyperbolicity and structure of smooth dynamical systems, In Proceedings of the International Congress of Mathematicians, (Warsaw, 1983), PWN. (1984), 1245–1253Google Scholar
Katok, A. and Mendoza, L.. Dynamical systems with nonuniformly hyperbolic behavior, In Katok, A. and Hasselblatt, B. Supplement to Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1995.10.1017/CBO9780511809187CrossRefGoogle Scholar
Meyer, K. R. and Sell, G. R.. An analytic proof of the shadowing lemma. Funkc. Ekvacioj. 30 (1987), 127–133.Google Scholar
Palmer, K. J.. Exponential dichotomies, the shadowing lemma and transversal homoclinic points. Dyn. Rep. 1 (1988), 265–306.Google Scholar
Palmer, K. J.. Shadowing in dynamical systems, theory and applications. Kluwer, Dordrecht, 2000.CrossRefGoogle Scholar
Pilyugin, S. Y.. Shadowing in Dynamical Systems. Lecture Notes in Mathmatics, Vol. 1706 (Springer, Berlin, 1999).Google Scholar
Rudin, W.. Principles of Mathematical Analysis. McGraw-Hill, New York, 1976.Google Scholar
Sinai, Y. G.. Gibbs measures in ergodic theory. Russian Math. Surveys. 27 (1972), 21–69.10.1070/RM1972v027n04ABEH001383CrossRefGoogle Scholar
Walters, P.. On the pseudo orbit tracing property and its relationship to stability. The Structure of Attractors in Dynamical Systems, Lecture Notes in Mathematics, Vol. 668, 231–244 (Springer, Berlin, 1978).10.1007/BFb0101795CrossRefGoogle Scholar
Zhou, L. and Zhang, W.. A projected discrete Gronwall’s inequality with sub-exponential growth. J. Differential Equa. Appl. 16 (2010), 931–943.10.1080/10236190802612873CrossRefGoogle Scholar