Hostname: page-component-5c6d5d7d68-ckgrl Total loading time: 0 Render date: 2024-08-07T07:09:01.461Z Has data issue: false hasContentIssue false
  • English
  • Français

A Finite-time Condition for Exponential Trichotomy in Infinite Dynamical Systems

Published online by Cambridge University Press:  20 November 2018

Arnaud Ducrot ,
Pierre Magal  and
Ousmane Seydi
Arnaud Ducrot
Affiliation:
Institut de Mathematiques de Bordeaux, Universite Bordeaux, 33000 Bordeaux, France. e-mail: arnaud.ducrot@u-bordeaux2.fr, pierre.magal@u-bordeaux2.fr, seydi86@hotmail.fr
Pierre Magal
Affiliation:
Institut de Mathematiques de Bordeaux, Universite Bordeaux, 33000 Bordeaux, France. e-mail: arnaud.ducrot@u-bordeaux2.fr, pierre.magal@u-bordeaux2.fr, seydi86@hotmail.fr
Ousmane Seydi
Affiliation:
Institut de Mathematiques de Bordeaux, Universite Bordeaux, 33000 Bordeaux, France. e-mail: arnaud.ducrot@u-bordeaux2.fr, pierre.magal@u-bordeaux2.fr, seydi86@hotmail.fr
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this article we study exponential trichotomy for infinite dimensional discrete time dynamical systems. The goal of this article is to prove that finite time exponential trichotomy conditions allow us to derive exponential trichotomy for arbitrary times. We present an application to the case of pseudo orbits in some neighborhood of a normally hyperbolic set.

Keywords

34D0934A10exponential trichotomyexponential dichotomydiscrete time dynamical systemsdifference equations
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 67 , Issue 5 , 01 October 2015 , pp. 1065 - 1090
Copyright
Copyright © Canadian Mathematical Society 2014

References

[1] Bates, P. W., Lu, K., and Zeng, C., Existence and persistence of invariant manifolds for semiflows in Banach space. Mem. Amer. Math. Soc. 135(1998), no. 645.Google Scholar
[2] Barreira, L. and Valls, C., Robustness of nonuniform exponential trichotomies in Banach spaces. J.Math. Anal. Appl. 351(2009), no. 1, 373–381.http://dx.doi.org/10.1016/j.jmaa.2008.10.030 Google Scholar
[3] Bochner, S., A new approach to almost periodicity. Proc. Nat. Acad. Sci. U.S.A. 48(1962), 2039–2043.http://dx.doi.org/10.1073/pnas.48.12.2039 Google Scholar
[4] Bohr, H., Almost periodic functions. Chelsea Publishing Company, New York, NY, 1947.Google Scholar
[5] chow, S.-N., Lin, X.-B., and Palmer, K. J., A shadowing lemma for maps in infinite dimensions. In: Differential equations (Xanthi, 1987), Lecture Notes in Pure and Appl. Math., 118, Dekker, New York, 1989, pp. 127–135.Google Scholar
[6] Coppel, W. A., Dichotomies in stability theory. Lecture Notes in Mathematics, 629, Springer-Verlag, Berlin-New York, 1978.Google Scholar
[7] Corduneanu, C., Almost periodic oscillations and waves. Springer, New York, 2009.Google Scholar
[8] Ducrot, A., Magal, P., and Seydi, O., Persistence of exponential trichotomy for linear operators: A Lyapunov-Perron approach. J. Dynam. Differential Equations, to appear.Google Scholar
[9] Fenichel, N., Persistence and smoothness of invariant manifolds for flows. Indiana Univ. Math. J. 21(1971/1972), 193–226.http://dx.doi.org/10.1512/iumj.1972.21.21017 Google Scholar
[10] Hale, J. K., Introduction to dynamic bifurcation. In: Bifurcation theory and applications (Montecatini, 1983), Lecture Notes in Math., 1057, Springer, Berlin, pp. 106–151.Google Scholar
[11] Hale, J. K. and Lin, X.-B., Heteroclinic orbits for retarded functional differential equations. J. Differential Equations 65(1986), no. 2, 175–202.http://dx.doi.org/10.1016/0022-0396(86)90032-X Google Scholar
[12] Henry, D., Geometric theory of semilinear parabolic equations. Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981.Google Scholar
[13] Hirsch, M. W., Pugh, C. C., and Shub, M., Invariant manifolds. Lecture Notes in Math., 583, Springer-Verlag, Berlin-New York, 1977.Google Scholar
[14] Minh, N. V., Géurékata, G. M. N’, and Yuan, R., Lectures on the asymptotic behavior of solutions of differential equations. Nova Science Publishers, Inc., New York, 2008.Google Scholar
[15] Palmer, K. J., A finite-time condition for exponential dichotomy. J. Difference Equ. Appl. 17(2011),no. 2, 221–234.http://dx.doi.org/10.1080/10236198.2010.549005 Google Scholar
[16] Palmer, K. J., A perturbation theorem for exponential dichotomies. Proc. Roy. Soc. Edinburgh Sect. A 106(1987), no. 1–2, 25–37. http://dx.doi.org/10.1017/S0308210500018175 Google Scholar
[17] Palmer, K. J., Exponential dichotomies for almost periodic equations. Proc. Amer. Math. Soc. 101(1987),no. 2, 293–298.http://dx.doi.org/10.1090/S0002-9939-1987-0902544-6 Google Scholar
[18] Palmer, K. J., Shadowing in dynamical systems. Theory and applications. Mathematics and its Applications, 501, Kluwer Academic Publishers, Dordrecht, 2000.Google Scholar
[19] Pliss, V. and Sell, G., Robustness of exponential dichotomies in infinite-dimensional dynamical systems. J. Dynam. Differential Equations 11(1999), no. 3, 471–513.http://dx.doi.org/10.1023/A:1021913903923 Google Scholar
[20] Pötzsche, C., Exponential dichotomies of linear dynamic equations on measure chains under slowly varying coefficients. J. Math. Anal. Appl. 289(2004), no. 1, 317–335.http://dx.doi.org/10.1016/j.jmaa.2003.09.063 Google Scholar
[21] Pötzsche, C., Geometric theory of discrete nonautonomous dynamical systems. Lecture Notes in Mathematics, 2002, Springer-Verlag, Berlin, 2010.Google Scholar
[22] Pötzsche, C., Bifurcations in nonautonomous dynamical systems: Results and tools in discrete time. In: Proceedings of the Workshop on Future Directions in Difference Equations, Colecc. Congr., 69, Univ. Vigo Serv. Publ., VIgo, Spain, 2011, pp. 163–212.Google Scholar
[23] Sacker, R. J. and Sell, G. R., A spectral theory for linear differential systems. J. Differential Equations 27(1978), no. 3, 320–358.http://dx.doi.org/10.1016/0022-0396(78)90057-8 Google Scholar
[24] Sakamoto, K., Invariant manifolds in singular perturbation problems for ordinary differential equations. Proc. Roy. Soc. Edinburgh Sect. A 116(1990), no. 1–2, 45–78.http://dx.doi.org/10.1017/S0308210500031371 Google Scholar
[25] Sakamoto, K., Estimates on the strength of exponential dichotomies and application to integral manifolds. J. Differential Equations 107(1994), no. 2, 259–279.http://dx.doi.org/10.1006/jdeq.1994.1012 Google Scholar
[26] Zhou, L., Lu, K., and Zhang, W., Roughness of tempered exponential dichotomies for infinite-dimensional random difference equations. J. Differential Equations 254(2013), no. 9, 4024–4046.http://dx.doi.org/10.1016/j.jde.2013.02.007 Google Scholar