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

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 $0<\gamma \le 1$. 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.

12 pages

References in corpus (2)

• Structure of characteristic Lyapunov vectors in spatiotemporal chaos
• Diverging fluctuations of the Lyapunov exponents

Cited by in corpus (1)

• Lyapunov Exponents for Temporal Networks