Velasco David, López Juan M, Pazó Diego
Instituto de Física de Cantabria (IFCA), CSIC-Universidad de Cantabria, 39005 Santander, Spain.
Phys Rev E. 2021 Sep;104(3-1):034216. doi: 10.1103/PhysRevE.104.034216.
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.
全局耦合映射（GCMs）是高维动力系统的典型例子。有趣的是，由一组弱耦合的相同混沌单元构成的GCMs通常会呈现出一种超混沌的“湍流”状态。十年前，竹内等人[《物理评论快报》107, 124101 (2011)PRLTAO0031 - 900710.1103/PhysRevLett.107.124101]提出理论，认为在湍流GCMs中，最大李雅普诺夫指数（LE），λ(N)，对系统规模N呈对数依赖关系：λ∞ - λ(N)≃c/lnN。我们重新审视这个问题，并通过解析和数值技术分析具有正乘数的湍流GCMs，结果表明与先前的预测相反，存在明显的非普适性。事实上，我们发现一种幂律标度关系λ∞ - λ(N)≃c/Nγ，其中γ是一个依赖于参数的指数，取值范围为0 < γ≤1。然而，对于强不相似的乘数，李雅普诺夫指数随N的变化更为缓慢，本文对此进行了数值研究。尽管我们的分析仅对具有正乘数的GCMs有效，但这表明在一般的GCMs中，不能想当然地认为李雅普诺夫指数存在一个普适的收敛规律。
