Asymptotic distribution of global errors in the numerical computations of dynamical systems

We propose an analysis of the effects introduced by finite-accuracy and round-off arithmetic on numerical computations of discrete dynamical systems. Our method, which uses the statistical tool of the decay of fidelity, computes the error by directly comparing the numerical orbit with the exact one (or, more precisely, with another numerical orbit computed with a much higher accuracy). Furthermore, as a model of the effects of round-off arithmetic on the map, we also consider a random perturbation of the exact orbit with an additive noise, for which exact results can be obtained for some prototype maps. We investigate the decay laws of fidelity and their relationship with the error probability distribution for regular and chaotic maps, for both additive and numerical noise. In particular, for regular maps we find an exponential decay for additive noise, and a power-law decay for numerical noise. For chaotic maps, numerical noise is equivalent to additive noise, and our method is suitable for identifying a threshold for the reliability of numerical results, i.e., the number of iterations below which global errors can be ignored. This threshold grows linearly with the number of bits used to represent real numbers.