Lyapunov exponents of linear stochastic systems

  • L. Arnold
  • W. Kliemann
  • E. Oeljeklaus
Part II: Linear Stochastic Systems. Stability Theory

DOI: 10.1007/BFb0076835

Part of the Lecture Notes in Mathematics book series (LNM, volume 1186)
Cite this paper as:
Arnold L., Kliemann W., Oeljeklaus E. (1986) Lyapunov exponents of linear stochastic systems. In: Arnold L., Wihstutz V. (eds) Lyapunov Exponents. Lecture Notes in Mathematics, vol 1186. Springer, Berlin, Heidelberg

Abstract

The asymptotic behavior of linear stochastic systems in Rd of the form ẋ = A(ξ(t))x, x(o) = xo ε Rd, ξ(t) stationary stochastic process, is investigated by means of geometric nonlinear control theory. Concerning the rotational behavior it is proved that the projection onto the unit sphere has a unique invariant probability. Concerning the stability it is proved that the solution x(t;xo) has an exponential growth rate
$$\lambda = \mathop {\lim }\limits_{t \to \infty } \tfrac{1}{t} \log |x(t;x_O )|$$
which is independent of chance and of xo and equal to the biggest Lyapunov exponent from the multiplicative ergodic theorem.

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Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • L. Arnold
    • 1
  • W. Kliemann
    • 2
  • E. Oeljeklaus
    • 3
  1. 1.Forschungsschwerpunkt Dynamische SystemeUniversitätBremen 33West Germany
  2. 2.Department of MathematicsIowa State UniversityAmesUSA
  3. 3.Fachbereich Mathematik/InformatikUniversitätBremen 33West Germany

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