Probability densities for the sums of iterates of the sine-circle map in the vicinity of the quasi-periodic edge of chaos

TL;DR

This paper studies the probability densities of sums of iterates in the sine-circle map near the quasi-periodic edge of chaos, revealing convergence to a q-Gaussian distribution at the onset of chaos.

Contribution

It provides numerical evidence that at the edge of chaos, the sum distributions deviate from the standard CLT and tend to a q-Gaussian form, highlighting non-ergodic behavior.

Findings

Probability densities approach a q-Gaussian near chaos onset

Standard CLT does not hold at the edge of chaos

Distribution convergence observed as golden mean is approached

Abstract

We investigate the probability density of rescaled sum of iterates of sine-circle map within quasi-periodic route to chaos. When the dynamical system is strongly mixing (i.e., ergodic), standard Central Limit Theorem (CLT) is expected to be valid, but at the edge of chaos where iterates have strong correlations, the standard CLT is not necessarily to be valid anymore. We discuss here the main characteristics of the central limit behavior of deterministic dynamical systems which exhibit quasi-periodic route to chaos. At the golden-mean onset of chaos for the sine-circle map, we numerically verify that the probability density appears to converge to a q-Gaussian with q<1 as the golden mean value is approached.