Lyapunov analysis captures the collective dynamics of large chaotic systems

TL;DR

This paper demonstrates that the collective behavior of large chaotic systems can be understood through Lyapunov spectra, revealing localized and delocalized modes and their relation to Perron-Frobenius dynamics, challenging traditional extensivity concepts.

Contribution

It introduces a Lyapunov-based framework to analyze collective dynamics in large chaotic systems, linking Lyapunov modes to Perron-Frobenius dynamics and revising the concept of extensivity.

Findings

Most Lyapunov modes are localized on few degrees of freedom.

Some Lyapunov modes are delocalized and act collectively.

A quantitative link exists between collective modes and Perron-Frobenius dynamics.

Abstract

We show, using generic globally-coupled systems, that the collective dynamics of large chaotic systems is encoded in their Lyapunov spectra: most modes are typically localized on a few degrees of freedom, but some are delocalized, acting collectively on the trajectory. For globally-coupled maps, we show moreover a quantitative correspondence between the collective modes and some of the so-called Perron-Frobenius dynamics. Our results imply that the conventional definition of extensivity must be changed as soon as collective dynamics sets in.