Advances in Operator Theory

General exponential dichotomies: from finite to infinite time

Luis Barreira and Claudia Valls

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

‎‎We consider exponential dichotomies on finite intervals and show that if the constants in the notion of an exponential dichotomy are chosen appropriately and uniformly on those intervals‎, ‎then there exists an exponential dichotomy on the whole line‎. ‎We consider the general case of a nonautonomous dynamics that need not be invertible‎. ‎Moreover‎, ‎we consider both cases of discrete and continuous time‎.

Article information

Source
Adv. Oper. Theory, Volume 4, Number 1 (2019), 215-225.

Dates
Received: 4 May 2018
Accepted: 27 June 2018
First available in Project Euclid: 7 July 2018

Permanent link to this document
https://projecteuclid.org/euclid.aot/1530928842

Digital Object Identifier
doi:10.15352/aot.1805-1364

Mathematical Reviews number (MathSciNet)
MR3867342

Zentralblatt MATH identifier
06946451

Subjects
Primary: ‎70F05
Secondary: ‎34D09

Keywords
exponential dichotomy ‎‎growth rate ‎nonautonomous dynamics

Citation

Barreira, Luis; Valls, Claudia. General exponential dichotomies: from finite to infinite time. Adv. Oper. Theory 4 (2019), no. 1, 215--225. doi:10.15352/aot.1805-1364. https://projecteuclid.org/euclid.aot/1530928842


Export citation

References

  • L. Barreira and C. Valls, Growth rates and nonuniform hyperbolicity, Discrete Contin. Dyn. Syst. 22 (2008), 509–528.
  • L. Barreira and C. Valls, Smooth robustness of parameterized perturbations of exponential dichotomies, J. Differential Equations 249 (2010), no. 8, 2021–2043.
  • L. Barreira and C. Valls, Robust nonuniform dichotomies and parameter dependence, J. Math. Anal. Appl. 373 (2011), 690–708.
  • S.-N. Chow and H. Leiva, Existence and roughness of the exponential dichotomy for skew-product semiflow in Banach spaces, J. Differential Equations 120 (1995), 429–477.
  • W. Coppel, Dichotomies and reducibility, J. Differential Equations 3 (1967), 500–521.
  • W. Coppel, Dichotomies in stability theory, Lect. Notes in Math. 629, Springer, 1978.
  • Ju. Dalec'kiĭ and M. Kreĭ n, Stability of solutions of differential equations in Banach space, Translations of Mathematical Monographs 43, Amer. Math. Soc., 1974.
  • A. Ducrot, P. Magal, and O. Seydi, A finite-time condition for exponential trichotomy in infinite dynamical systems, Canad. J. Math. 67 (2015), 1065–1090.
  • A. Ducrot, P. Magal, and O. Seydi, Persistence of exponential trichotomy for linear operators: a Lyapunov–Perron approach, J. Dynam. Differential Equations 28 (2016), 93–126.
  • J. Hale, Asymptotic behavior of dissipative systems, Mathematical Surveys and Monographs 25, Amer. Math. Soc., 1988.
  • D. Henry, Geometric theory of semilinear parabolic equations, Lect. Notes in Math. 840, Springer, 1981.
  • D. Henry, Exponential dichotomies, the shadowing lemma and homoclinic orbits in Banach spaces, Resenhas 1 (1994), 381–401.
  • J. Massera and J. Schäffer, Linear differential equations and functional analysis. I, Ann. of Math. (2) 67 (1958), 517–573.
  • J. Massera and J. Schäffer, Linear Differential Equations and Function Spaces, Pure and Applied Mathematics 21, Academic Press, 1966.
  • R. Naulin and M. Pinto, Admissible perturbations of exponential dichotomy roughness, Nonlinear Anal. 31 (1998), 559–571.
  • V. Oseledets, A multiplicative ergodic theorem. Liapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc. 19 (1968), 197–221.
  • K. Palmer, Shadowing in Dynamical Systems. Theory and Applications, Mathematics and its Applications 501, Kluwer Academic Publishers, 2000.
  • K. Palmer, A finite-time condition for exponential dichotomy, J. Difference. Equ. Appl. 17 (2011), 221–234.
  • O. Perron, Die Stabilitätsfrage bei Differentialgleichungen, Math. Z. 32 (1930), 703–728.
  • Ya. Pesin, Families of invariant manifolds corresponding to nonzero characteristic exponents, Math. USSR-Izv. 10 (1976), 1261–1305.
  • Ya. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory, Russian Math. Surveys 32 (1977), 55–114.
  • Ya. Pesin, Geodesic flows on closed Riemannian manifolds without focal points, Math. USSR-Izv. 11 (1977), 1195–1228.
  • V. Pliss and G. Sell, Robustness of exponential dichotomies in infinite-dimensional dynamical systems, J. Dynam. Differential Equations 11 (1999), 471–513.
  • L. Popescu, Exponential dichotomy roughness on Banach spaces, J. Math. Anal. Appl. 314 (2006), 436–454.
  • G. Sell and Y. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences 143, Springer, 2002.