Extensive and Sub-Extensive Chaos in Globally-Coupled Dynamical Systems

TL;DR

This paper investigates chaos in globally-coupled dynamical systems, revealing that their Lyapunov spectra exhibit both extensive and sub-extensive behaviors, with implications for understanding complex collective dynamics.

Contribution

The study combines analytical and numerical methods to characterize the Lyapunov spectrum in globally-coupled systems, identifying the coexistence of extensive and sub-extensive chaos.

Findings

Lyapunov spectrum is asymptotically flat at λ₀ for large N.

Sub-extensive bands contain about log N exponents with specific scaling.

Chaos exhibits both extensive and sub-extensive features depending on the spectrum region.

Abstract

Using a combination of analytical and numerical techniques, we show that chaos in globally-coupled identical dynamical systems, be they dissipative or Hamiltonian, is both extensive and sub-extensive: their spectrum of Lyapunov exponents is asymptotically flat (thus extensive) at the value $\lambda_0$ given by a single unit forced by the mean-field, but sandwiched between sub-extensive bands containing typically $\mathcal{O}(\log N)$ exponents whose values vary as $\lambda \simeq \lambda_\infty + c/\log N$ with $\lambda_\infty \neq \lambda_0$.