Error distribution in randomly perturbed orbits.

Given an observable f defined on the phase space of some dynamical system generated by the mapT, we consider the error between the value of the function f(Tnx0) computed at time n along the orbit with initial condition x0, and the value f(Tn x0) of the same observable computed by replacing the map Tn with the composition of maps T(omega)(n)o...oT(omega)1, where each T(omega) is chosen randomly, by varying omega, in a neighborhood of size of T. We show that the random variable Delta(n)omega [triple bond] f(Tnx0) -f(T(omega)nx0), 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.