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)PRLTAO0031-900710.1103/PhysRevLett.107.124101] theorized that in turbulent GCMs the largest Lyapunov exponent (LE), λ(N), depends logarithmically on the system size N: λ_{∞}-λ(N)≃c/lnN. 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 λ_{∞}-λ(N)≃c/N^{γ}, where γ is a parameter-dependent exponent in the range 0<γ≤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.