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Globally coupled maps (GCMs) are prototypical examples of high-dimensionaldynamical systems. Interestingly, GCMs formed by an ensemble of weakly coupledidentical chaotic units generically exhibit a hyperchaotic 'turbulent' state. Adecade ago, Takeuchi et al. [Phys. Rev. Lett. 107, 124101 (2011)] theorizedthat 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 positivemultipliers to show that there is a remarkable lack of universality, inconflict with the previous prediction. In fact, we find a power-law scaling$\lambda_\infty-\lambda(N)\simeq c/N^{\gamma}$, where $\gamma$ is aparameter-dependent exponent in the range $0<\gamma\le1$. However, for stronglydissimilar multipliers, the LE varies with $N$ in a slower fashion, which ishere numerically explored. Although our analysis is only valid for GCMs withpositive multipliers, it suggests that a universal convergence law for the LEcannot be taken for granted in general GCMs.

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