Central limit behavior of deterministic dynamical systems

TL;DR

This paper studies the statistical behavior of sums of deterministic dynamical system iterates, showing how the classical CLT applies or fails at critical points, and introduces a generalized CLT involving q-Gaussians.

Contribution

It analytically derives corrections to the CLT for certain maps and provides numerical evidence for a q-Gaussian limit at criticality, revealing universal behavior.

Findings

CLT holds for mixing systems with finite iterates

At critical points, CLT fails due to strong correlations

Probability densities converge to q-Gaussians at criticality

Abstract

We investigate the probability density of rescaled sums of iterates of deterministic dynamical systems, a problem relevant for many complex physical systems consisting of dependent random variables. A Central Limit Theorem (CLT) is only valid if the dynamical system under consideration is sufficiently mixing. For the fully developed logistic map and a cubic map we analytically calculate the leading-order corrections to the CLT if only a finite number of iterates is added and rescaled, and find excellent agreement with numerical experiments. At the critical point of period doubling accumulation, a CLT is not valid anymore due to strong temporal correlations between the iterates. Nevertheless, we provide numerical evidence that in this case the probability density converges to a $q$-Gaussian, thus leading to a power-law generalization of the CLT. The above behavior is universal and independent of the order of the maximum of the map considered, i.e. relevant for large classes of critical dynamical systems.