Error distribution in randomly perturbed orbits

Abstract

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(T^n x0) computed at time n along the orbit with initial condition x0 , and the value f(T^n_omega x0) of the same observable computed by replacing the map T n with the composition of maps T_omegai , where each T_omega is chosen randomly, by varying omega, in a neighborhood of size epsilon of T. We show that the random variable Delta^epsilon_n=f(T^n x0)-f(T_omega^n x0), depending on the initial condition x0 and on the choice of the realization omega , will converge in distribution when n → infinity 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.