Relaxation to the asymptotic distribution of global errors due to round off

¹  Department of Physics, University of Bologna - Bologna, Italy, EU
²  UMR-6207 Centre de Physique Théorique, CNRS, Universités d'Aix-Marseille I, II, Université du Sud, Toulon-Var and FRUMAM - Toulon, France, EU

Corresponding author: francesco.zanlungo@gmail.com

Received: 16 November 2009
Accepted: 1 February 2010

Abstract

We propose an analysis of the effects introduced by finite accuracy and round-off arithmetic on discrete dynamical systems. We investigate, from a statistical viewpoint and using the tool of the decay of fidelity, the error of the numerical orbit with respect to the exact one. As a model we consider a random perturbation of the exact orbit with an additive noise, for which exact results can be obtained for some prototype maps. For regular anysocrounous maps the fidelity has a power law decay, whereas the decay is exponential if a random perturbation is introduced. For chaotic maps the decay is superexponential after an initial plateau and our method is suitable to identify the reliability threshold of numerical results, i.e. a number of iterations below which global errors can be ignored. The same behaviour is observed if a random perturbation is introduced.