Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-28T12:24:02.539Z Has data issue: false hasContentIssue false
  • English
  • Français

Structural stability of linear random dynamical systems

Published online by Cambridge University Press:  14 October 2010

Nguyen Dinh Cong
Nguyen Dinh Cong
Affiliation:
Institut für Dynamische Systeme, Universität Bremen, Postfach 330 440, 28334 Bremen, Germany
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, structural stability of discrete-time linear random dynamical systems is studied. A random dynamical system is called structurally stable with respect to a random norm if it is topologically conjugate to any random dynamical system which is sufficiently close to it in this norm. We prove that a discrete-time linear random dynamical system is structurally stable with respect to its Lyapunov norms if and only if it is hyperbolic.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 16 , Issue 6 , December 1996 , pp. 1207 - 1220
Copyright
Copyright © Cambridge University Press 1996

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

REFERENCES

[Arn] Arnold, L.. Random Dynamical Systems. Preliminary version 2, Bremen, 1994.Google Scholar
[AC] Arnold, L. and Crauel, H.. Random dynamical systems. Lyapunov Exponents, Oberwolfach 1990 (Lecture Notes in Mathematics 1486). Eds Arnold, L., Crauel, H. and Eckmann, J.-P.. Springer, Berlin, 1991, pp. 1–22.Google Scholar
[Box] Boxler, P.. A stochastic version of center manifold theory. Probab. Th. Rel. Fields 83 (1989), 509–545.CrossRefGoogle Scholar
[C] Cong, Nguyen Dinh. Topological classification of linear hyperbolic cocycles. J. Dyn. Differential Equations. To appear.Google Scholar
[Gun] Gundlach, V. M.. Random homoclinic orbits. Random & Computational Dynamics 3 (1995), 1–33.Google Scholar
[Irw] Irwin, M. C.. Smooth Dynamical Systems. Academic, London, 1980.Google Scholar
[Joh] Johnson, R. A.. Remarks on linear differential systems with measurable coefficients. Proc. Amer. Math. Soc. 100 (1987), 491–504.CrossRefGoogle Scholar
[Kn] Knill, O.. The upper Lyapunov exponent of Sl(2, R) cocycles: discontinuity and the problem of positivity. Lyapunov Exponents, Oberwolfach 1990 (Lecture Notes in Mathematics I486). Eds Arnold, L., Crauel, H. and Eckmann, J.-P.. Springer, Berlin, 1991, pp. 86–97.Google Scholar
[Os] Oseledets, V. I.. A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc. 19 (1968), 197–231.Google Scholar
[Pal] Palmer, K. J.. Exponential dichotomies, the shadowing lemma and transversal homoclinic points. Dynamics Reported 1 (1988), 265–306.CrossRefGoogle Scholar
[Rob] Robbin, J. W.. Topological conjugacy and structural stability for discrete dynamical systems. Bull. Amer. Math. Soc. 78 (1972), 923–952.CrossRefGoogle Scholar