Lyapunov exponents of nilpotent ito systems

On the tangent flow of a stochastic differential equation with fast drift

“…ThenĀ y is a hypoelliptic operator on S 1 for each y ∈ R (see [1] or [13] or the calculations of Section 5). Thus, for each y ∈ R there is a uniqueμ y ∈ P(S 1 ) such that …”

“…The Khasminskii-Furstenburg formula allows us to study the evolution of Ξ ε,x,y t V . To write all of this out, we adopt most of the notation of [13]. Define S 1 def = {x ∈ R 2 : x = 1} and let ·, · denote the standard inner product in R 2 .…”

“…We start by using the transformation of Pinsky and Wihstutz [13] to identify the dominant terms of (3). To help clarify things, let's set up some notation.…”

Stability of noisy nonlinear auto-parametric systems

“…The idea behind such transformations is to use instead of the coordinates which remains parallel to (x, y), a new coordinate system (u, v) with one axis u moving so as to remain tangent to the unperturbed trajectory, while the other axis v remains perpendicular to the unperturbed trajectory. In Section 4.2, due to the nilpotent structure of the linear variational equations, the Pinsky and Wihstutz [13] rescaling is used in the linear variational equations to derive the Furstenberg-Khasminskii formula. In Sections 4.3 and 4.4 we appeal to the results of Sri Namachchivaya and Van Roessel [14] and Imkeller and Lederer [15] to evaluate the first term in the asymptotic expansion of the top Lyapunov exponent.…”

Lyapunov exponents indicate stability and detect stochastic bifurcations

“…The real noise case is treated in Arnold~ Papanicolaou and Wihstutz [7], the white noise case in Pardoux and Wihstutz [25] and in Pinsky and Wihstutz [26], and the case of dichotomic noise (i.e. which can only take two different values) in Arnold and Kloeden [5].…”