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Detailed view Article in peer-reviewed journal
Chaos 19 (2009) 043118 (2009)
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Error distribution in randomly perturbed orbits
Sandro Vaienti1, 2, Giorgio Turchetti3, Francesco Zanlungo3, Philippe Marie1

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(Tnx0) computed at time n along the orbit with initial condition x0, and the value f(Tωnx0) of the same observable computed by replacing the map Tn with the composition of maps Tωn∘⋯∘Tω1, 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(Tnx0)−f(Tωnx0), 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.
1:  CPT - Centre de Physique Théorique
2:  FRUMAM - Fédération de Recherche des Unités de MAthématiques de Marseille
3:  Department of Physics, University of Bologna