Central Limit Violation in the Kuramoto model at the ’Edge of Chaos’

We study the relationship between chaotic behavior and the violation of the Central Limit Theorem (CLT) in the Kuramoto model. We calculate sums of angles at equidistant times along deterministic trajectories of single oscillators and we show that, when chaos is sufficiently strong , the Pdfs of the sums tend to a Gaussian, consistently with the standard CLT. On the other hand, when the system is at the ”edge of chaos” (i.e. in a regime with vanishing Lyapunov exponents), robust q-Gaussian-like attractors naturally emerge, consistently with recently proved generalizations of the CLT. PACS. 64.60.My Phase transitions – 89.75.-k Chaos – 89.75.-k Long-range interacting systems The relationship between very weak chaos and the violation of the Central Limit Theorem (CLT) has been recently explored by several authors, both in low-dimensional chaotic maps [1,2,3,4] and in long-range many-body systems [5,6]. The Standard Central Limit Theorem (CLT) states that the (rescaled) sum of n independent random variables with finite variance has a Gaussian distribution in the limit n → ∞. On the other hand, the theorem does not hold when the variables under consideration are weakly mixing and strongly correlated, as it happens in systems with zero or vanishing Lyapunov exponents, i.e. systems at the ”edge of chaos”. In order to consider this case, it has recently been introduced a q-generalization of the standard CLT, which states that for certain classes of strongly correlated random variables the (rescaled) sum approaches a q-Gaussian distribution [7]. The latter is a generalization of a Gaussian distribution which emerges in the context of the nonextensive statistical mechanics [8]. It is defined as Gq(β, x) = A(q, β)[1 − (1 − q)βx2]1/(1−q), where q is the entropic index (such that for q = 1 one recovers the usual Gaussian), β a parameter related to the width of the distribution and A(q, β) is a normalizing constant. In previous works [5,6] we showed that q-Gaussian attractors exist for the rescaled sums of the angular velocities, calculated along deterministic trajectories of single oscillators (time-average Pdfs), in the weakly chaotic quasistationary (QSS) regime of the Hamiltonian Mean Field (HMF) model [9]. On the other hand, we also pointed out [10,11] several analogies (metastability, continuous phase transition, chaotic behavior) between the HMF model and the Kuramoto model [12], probably due to their common origin from a more general model of coupled drivendamped pendula. In this paper we present new numerical results which extend these analogies to the CLT features. In particular we will show that also in the context of the Kuramoto model a very weakly chaotic microscopic dynamics with vanishing Lyapunov exponents gives rise to the emergence of robust q-Gaussian-like attractors, which disappear when the level of chaos increases, thus restoring the standard CLT behavior. In the first part of the paper we describe the Kuramoto model phase transition scenario, with specific attention to the chaotic behavior and to the role played by the distribution of the natural frequencies (uniform or Gaussian). In the second part we discuss the numerical results on the CLT behavior, which are strongly influenced by the interplay between chaos and synchronization. 1 Phase transitions, Chaos and Metastability in the Kuramoto model The Kuramoto model [12,13,14] describes the following dissipative dynamics of a system of N sinusoidally coupled oscillators: