Abstract
For a one-sided nonautonomous dynamics defined by a sequence of invertible matrices, we develop a spectral theory (in the sense of Sacker and Sell) for the notion of a nonuniform exponential dichotomy with an arbitrarily small nonuniform part. We emphasize that this notion is ubiquitous in the context of ergodic theory, unlike the notion of a uniform exponential dichotomy. In particular, we show that each Lyapunov exponent belongs to one interval of the spectrum. We also consider a class of sufficiently small nonlinear perturbations of a linear dynamics satisfying a nonuniform bounded growth condition and we show that each solution is either eventually zero or the Lyapunov exponents belong to one interval of the spectrum.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aulbach, B., Siegmund, S.: The dichotomy spectrum for noninvertible systems of linear difference equations. J. Differ. Equ. Appl. 7, 895–913 (2001)
Aulbach, B., Siegmund, S.: A spectral theory for nonautonomous difference equations, In: New Trends in Difference Equations (Temuco, 2000), pp. 45–55. Taylor & Francis (2002)
Barreira, L., Pesin, Y.: Nonuniform Hyperbolicity, Encyclopedia of Mathematics and its Applications, vol. 115. Cambridge University Press, Cambridge (2007)
Barreira, L., Valls, C.: Stability theory and Lyapunov regularity. J. Differ. Equ. 232, 675–701 (2007)
Barreira, L., Valls, C.: A Perron-type theorem for nonautonomous differential equations. J. Differ. Equ. 258, 339–361 (2015)
Coffman, C.: Asymptotic behavior of solutions of ordinary difference equations. Trans. Am. Math. Soc. 110, 22–51 (1964)
Coppel, W.: Stability and Asymptotic Behavior of Differential Equations. D. C. Heath and Co., Lexington (1965)
Hartman, P., Wintner, A.: Asymptotic integrations of linear differential equations. Am. J. Math. 77, 45–86 (1955)
Lettenmeyer, F.: Über das asymptotische Verhalten der Lösungen von Differentialgleichungen und Differentialgleichungssystemen, Verlag d. Bayr. Akad. d. Wiss., pp. 201–252 (1929)
Matsui, K., Matsunaga, H., Murakami, S.: Perron type theorem for functional differential equations with infinite delay in a Banach space. Nonlinear Anal. 69, 3821–3837 (2008)
Perron, O.: Über Stabilität und asymptotisches Verhalten der Integrale von Differentialgleichungssystemen. Math. Z. 29, 129–160 (1929)
Pituk, M.: Asymptotic behavior and oscillation of functional differential equations. J. Math. Anal. Appl. 322, 1140–1158 (2006)
Pituk, M.: A Perron type theorem for functional differential equations. J. Math. Anal. Appl. 316, 24–41 (2006)
Sacker, R., Sell, G.: A spectral theory for linear differential systems. J. Differ. Equ. 27, 320–358 (1978)
Siegmund, S.: Dichotomy spectrum for nonautonomous differential equations. J. Dyn. Differ. Equ. 14, 243–258 (2002)
Author information
Authors and Affiliations
Corresponding author
Additional information
L. B. and C. V. were supported by FCT/Portugal through UID/MAT/04459/2013. D. D. was supported by an Australian Research Council Discovery Project DP150100017 and Croatian Science Foundation under the project IP-2014-09-2285.
Rights and permissions
About this article
Cite this article
Barreira, L., Dragičević, D. & Valls, C. Nonuniform Spectrum on the Half Line and Perturbations. Results Math 72, 125–143 (2017). https://doi.org/10.1007/s00025-016-0626-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00025-016-0626-8