Local Lyapunov exponents

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 A ∈ C(ℝ^d,K^n×n) is a continuous matrix function (K = ℝ or ℂ), d ∈ ℕ and n ∈ ℕ are the dimensions of the state spaces of X^ε and Z^ε, respectively, ε ≥ 0 parametrizes the intensity of (W _t )_t≥0 which denotes a Brownian motion in ℝ^d on a complete probability space (Ω,F, ℙ) and X^ε,ξ is a diffusion starting in ξ ∈ ℝ^d, defined by the SDE (2.1) such that the assumptions 2.1.1 hold. For Z^ε, 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 Z^ε.