Central Limit Behavior in the Kuramoto model at the 'Edge of Chaos'

TL;DR

This paper investigates how the Central Limit Theorem manifests in the Kuramoto model, showing Gaussian behavior in chaotic regimes and q-Gaussian attractors at the edge of chaos, linking chaos strength to distribution types.

Contribution

It demonstrates the transition from Gaussian to q-Gaussian distributions in the Kuramoto model depending on the chaos level, highlighting the role of Lyapunov exponents.

Findings

Gaussian PDFs in strongly chaotic regimes

q-Gaussian attractors at the edge of chaos

Distribution type correlates with chaos intensity

Abstract

We study the relationship between chaotic behavior and the Central Limit Theorem (CLT) in the Kuramoto model. We calculate sums of angles at equidistant times along deterministic trajectories of single oscillators and we show that, when chaos is sufficiently strong, the Pdfs of the sums tend to a Gaussian, consistently with the standard CLT. On the other hand, when the system is at the "edge of chaos" (i.e. in a regime with vanishing Lyapunov exponents), robust $q$-Gaussian-like attractors naturally emerge, consistently with recently proved generalizations of the CLT.