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 (Tωn 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 Tω is chosen randomly, by varying ω, in a neighborhood of size ε of T. We show that the random variable Δnε ≡ f (Tn x 0) -f (Tωn 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.
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics
- Physics and Astronomy(all)
- Applied Mathematics