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Lyapunov exponents of linear stochastic systems

Part II: Linear Stochastic Systems. Stability Theory

Part of the Lecture Notes in Mathematics book series (LNM,volume 1186)

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.

Keywords

  • Lyapunov Exponent
  • Stochastic System
  • Principal Eigenvalue
  • Integral Manifold
  • Exponential Growth Rate

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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. https://doi.org/10.1007/BFb0076835

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  • DOI: https://doi.org/10.1007/BFb0076835

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