Nonuniversal large-size asymptotics of the Lyapunov exponent in turbulent globally coupled maps

TL;DR

This paper investigates the asymptotic behavior of the largest Lyapunov exponent in turbulent globally coupled maps, revealing a nonuniversal power-law scaling with system size that challenges previous logarithmic predictions.

Contribution

It demonstrates that the convergence of the Lyapunov exponent in turbulent GCMs is nonuniversal and depends on system parameters, contrasting earlier theoretical predictions.

Findings

Lyapunov exponent scales as a power law with system size

Scaling exponent varies with system parameters

Universal convergence law for Lyapunov exponent is unlikely

Abstract

Globally coupled maps (GCMs) are prototypical examples of high-dimensional dynamical systems. Interestingly, GCMs formed by an ensemble of weakly coupled identical chaotic units generically exhibit a hyperchaotic 'turbulent' state. A decade ago, Takeuchi et al. [Phys. Rev. Lett. 107, 124101 (2011)] theorized that in turbulent GCMs the largest Lyapunov exponent (LE), $\lambda(N)$, depends logarithmically on the system size $N$: $\lambda_\infty-\lambda(N)\simeq c/\ln N$. We revisit the problem and analyze, by means of analytical and numerical techniques, turbulent GCMs with positive multipliers to show that there is a remarkable lack of universality, in conflict with the previous prediction. In fact, we find a power-law scaling $\lambda_\infty-\lambda(N)\simeq c/N^{\gamma}$, where $\gamma$ is a parameter-dependent exponent in the range $0<\gamma\le1$. However, for strongly dissimilar multipliers, the LE varies with $N$ in a slower fashion, which is here numerically explored. Although our analysis is only valid for GCMs with positive multipliers, it suggests that a universal convergence law for the LE cannot be taken for granted in general GCMs.