Error distribution in randomly perturbed orbits

Sandro Vaienti (1, 2) , Giorgio Turchetti (3) , Francesco Zanlungo (3) , Philippe Marie (2)

1 CPT - Centre de Physique Théorique - UMR 6207
2 CPT - Centre de Physique Théorique - UMR 7332
3 UNIBO - Alma Mater Studiorum Università di Bologna = University of Bologna

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.

Domaines: Systèmes dynamiques [math.DS]

Sandro Vaienti, Giorgio Turchetti, Francesco Zanlungo, Philippe Marie. Error distribution in randomly perturbed orbits. Chaos: An Interdisciplinary Journal of Nonlinear Science, 2009, 19, pp.043118 (2009). ⟨10.1063/1.3267510⟩. ⟨hal-00476224⟩