Lyapunov exponents: A survey

Stability of a Stochastic Two-Dimensional Non-Hamiltonian System

“…No theorem has had so direct and powerful an influence upon the study of stochastic stability of noisy dynamical systems as the multiplicative ergodic theorem (MET) of Oseledets [1], which established the existence of (typically) finitely many deterministic exponential growth rates called Lyapunov exponents. The stability of linear stochastic systems based on the MET has been well established [2,3] and the top Lyapunov exponent can be evaluated explicitly with relative ease when the noisy perturbations and dissipation are weak [4,5]. The challenge has been to extend the existing techniques in order to explicitly evaluate the top Lyapunov exponent of nonlinear systems with noise, and in particular additive white noise.…”

Pathwise Stability of Degenerate Stochastic Evolutions

“…IEOT For more results on finite dimensional stochastic Lyapunov exponents, see the overview paper [4], the book by Khasminskii [22], and e.g. [2,3,24,31] and [33].…”

Lyapunov exponents, path-integrals and forms

“…For stochastic systems various inequivalent versions of Lyapunov-like quantitites have been studied in the literature [6] . Here we only mention the one employed by Benzi et al [8] and by Graham [9] .…”

“…For deterministic systems the Lyapunov exponents are computed from the properties of a single trajectory, i.e., they are labelled by the initial point of the respective trajectory. More recently the concept of Lyapunov exponents has been generalized also to stochastic systems [6] . In this case the Lyapunov exponents, like all observables, are obtained by averaging over (infinitely) many trajectories.…”