Nonuniversal Large-Size Asymptotics Of The Lyapunov Exponent In Turbulent Globally Coupled Maps

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(infinity) - lambda(N)similar or equal to 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(infinity) - lambda(N) similar or equal to c/N-gamma , where gamma is a parameter-dependent exponent in the range 0 < gamma <= 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.