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Local Lyapunov exponents

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1963)

In this chapter the goal of obtaining the “local Lyapunov exponents” as sublimiting exponential growth rates is tackled. As already described, the system under consideration is the real-noise driven stochastic system

$$dZ^{\varepsilon}_t = {\bf A} (X^{\varepsilon}_t) \; Z^{\varepsilon}_t dt \; dX{^\varepsilon}_{\!\!\!\!t} = b (X^{\varepsilon}_t)\; dt + \sqrt{\varepsilon} \; \sigma \; (X^{\varepsilon}_t) \; dW_t$$
((1))

where AC(ℝd,Kn×n) is a continuous matrix function (K = ℝ or ℂ), d ∈ ℕ and n ∈ ℕ are the dimensions of the state spaces of Xε and Zε, respectively, ε ≥ 0 parametrizes the intensity of (W t )t≥0 which denotes a Brownian motion in ℝd on a complete probability space (Ω,F, ℙ) and Xε,ξ is a diffusion starting in ξ ∈ ℝd, defined by the SDE (2.1) such that the assumptions 2.1.1 hold. For Zε, solving the random vector differential equation

$$dZ^{\varepsilon}_t = {\bf A}( X^{\varepsilon, x}_t (\omega)) \; Z^{\varepsilon}_t \; dt, \; \; \; Z^{\varepsilon}_0 = z \in {\rm K}^n$$

we will use the equivalent notations

$$Z^{\varepsilon} : {\rm R}_+ \times \Omega \times {\rm R}^d \times {\rm K}^n \rightarrow {\rm K}^n$$
$$(t,\omega,x,z) \mapsto Z^{\varepsilon}(t,\omega,x,z) \equiv Z^{\varepsilon}(t,\omega,x)z \equiv Z^{\varepsilon}_t(\omega,x)z \equiv Z^{\varepsilon}_t(\omega,x,z)$$

as before, where

$$Z^{\varepsilon}_t(\omega,x) \equiv Z^{\varepsilon} (t,\omega,x, .)$$

solves the random matrix differential equation

$$dZ^{\varepsilon}_t = {\bf A}( X^{\varepsilon, x}_t (\omega))\; Z^{\varepsilon}_t \;dt, \; \; \; Z^{\varepsilon}_0 = {\rm id}_{{\rm K}^n}$$

The object of interest is the exponential growth rate

$${1 \over T({\varepsilon})} \;{\rm log} \mid Z_{T({\varepsilon})}^{\varepsilon} (\omega, x, z)\mid$$

on the time scale T(ε). Any limit as ε → 0 of this rate will be called local Lyapunov exponent of Zε.

Keywords

  • Lyapunov Exponent
  • Sojourn Time
  • Exit Time
  • Real Noise
  • 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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© 2009 Springer-Verlag Berlin Heidelberg

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(2009). Local Lyapunov exponents. In: Local Lyapunov Exponents. Lecture Notes in Mathematics, vol 1963. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85964-2_4

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