Extreme Value laws for dynamical systems under observational noise

TL;DR

This paper demonstrates that extreme value laws hold for dynamical systems affected by observational noise, showing that noise preserves attractor structure and affects the statistical properties of observed orbits.

Contribution

It proves the existence of extreme value laws under observational noise and provides numerical evidence on how noise influences these laws and the attractor structure.

Findings

Extreme value laws hold under observational noise.

Observational noise preserves the attractor's structure.

Noise influences the magnitude of deviations from deterministic laws.

Abstract

In this paper we prove the existence of Extreme Value Laws for dynamical systems perturbed by 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 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 any fractal structures present on the attractor.