Error distribution in randomly perturbed orbits

Given an observable f defined on the phase space of some dynamical system generated by the map T, we consider the error between the value of the function f (Tⁿ x₀) computed at time n along the orbit with initial condition x₀, and the value f (T_ωⁿ x ₀) of the same observable computed by replacing the map Tⁿ with the composition of maps T _ωn T o⋯oT _ω1, where each T_ω is chosen randomly, by varying ω, in a neighborhood of size ε of T. We show that the random variable Δ_n^ε ≡ f (Tⁿ x ₀) -f (T_ωⁿ x₀), depending on the initial condition x₀ and on the choice of the realization ω, will converge in distribution when n→∞ to what we call the asymptotic error. We study in detail the density of the distribution function of the asymptotic error for a wide class of dynamical systems perturbed with additive noise: for a few of them we give rigorous results, for the others we provide a numerical investigation. Our study is intended as a model for the effects of numerical noise due to roundoff on dynamical systems.