Extreme value laws for dynamical systems under observational noise

Abstract

In this paper we prove the existence of extreme value laws for dynamical systems perturbed by the instrument-like-error, also called observational noise. An orbit perturbed with observational noise mimics the behavior of an instrumentally recorded time series. Instrument characteristics–defined as precision and accuracy–act both by truncating and randomly displacing the real value of a measured observable. Here we analyze both these effects from a theoretical and a numerical point of view. First we show that classical extreme value laws can be found for orbits of dynamical systems perturbed with observational noise. Then we present numerical experiments to support the theoretical findings and give an indication of the order of magnitude of the instrumental perturbations which cause relevant deviations from the extreme value laws observed in deterministic dynamical systems. Finally, we show that the observational noise preserves the structure of the deterministic attractor. This goes against the common assumption that random transformations cause the orbits asymptotically fill the ambient space with a loss of information about the fractal structure of the attractor.