Physica D 272 (2014) 18–25 Contents lists available at ScienceDirect Physica D journal homepage: www.elsevier.com/locate/physd Relationships and scaling laws among correlation, fractality, Lyapunov divergence and q-Gaussian distributions Ozgur Afsar a , Ugur Tirnakli a,b,∗ a Department of Physics, Faculty of Science, Ege University, 35100 Izmir, Turkey b Division of Statistical Mechanics and Complexity, Institute of Theoretical and Applied Physics (ITAP) Kaygiseki Mevkii, 48740 Turunc, Mugla, Turkey highlights • First establishment of relationships among correlation, fractality, Lyapunov divergence and q-Gaussians. • Novel critical exponents for these relationships are obtained. • All scaling relations obtained here are shown to be consistent with the universal Huberman–Rudnick scaling law. • New evidence supporting the central limit behaviour at the chaos threshold is given by a q-Gaussian. article info Article history: Received 17 September 2013 Received in revised form 16 January 2014 Accepted 17 January 2014 Available online 27 January 2014 Communicated by T. Wanner Keywords: Dissipative maps Chaos threshold Numerical simulations of chaotic systems Nonextensive statistical mechanics Central limit behaviour abstract We numerically introduce the relationships among correlation, fractality, Lyapunov divergence and q- Gaussian distributions. The scaling arguments between the range of the q-Gaussian and correlation, fractality, Lyapunov divergence are obtained for periodic windows (i.e., periods 2, 3 and 5) of the logistic map as chaos threshold is approached. Firstly, we show that the range of the q-Gaussian (g ) tends to infinity as the measure of the deviation from the correlation dimension (D corr = 0.5) at the chaos threshold, (this deviation will be denoted by l), approaches to zero. Moreover, we verify that a scaling law of type 1/g ∝ l τ is evident with the critical exponent τ = 0.23 ± 0.01. Similarly, as chaos threshold is approached, the quantity l scales as l ∝ (a − a c ) γ , where the exponent is γ = 0.84 ± 0.01. Secondly, we also show that the range of the q-Gaussian exhibits a scaling law with the correlation length (1/g ∝ ξ −µ ), Lyapunov divergence (1/g ∝ λ µ ) and the distance to the critical box counting fractal dimension (1/g ∝ (D − D c ) µ ) with the same exponent µ ∼ = 0.43. Finally, we numerically verify that these three quantities (ξ , λ, D − D c ) scale with the distance to the critical control parameter of the map (i.e., a − a c ) in accordance with the universal Huberman–Rudnick scaling law with the same exponent ν = 0.448 ± 0.003. All these findings can be considered as a new evidence supporting that the central limit behaviour at the chaos threshold is given by a q-Gaussian. © 2014 Elsevier B.V. All rights reserved. 1. Introduction Random variables from many complex systems in nature, such as financial data [1], earthquakes [2], long-range interacting many body systems [3] and from the synthetic data of many mathematical models like cubic map, logistic map, logarithmic map, standard sine-circle map [4–9], Kuramoto model [10], conservative maps [11] in the literature seem to exhibit q-Gaussian behaviour for the appropriate probability distribution functions in some particular points or regions. Although it is well known that long-range correlations and fractal nature are common properties ∗ Corresponding author at: Department of Physics, Faculty of Science, Ege University, 35100 Izmir, Turkey. Tel.: +90 2323111768. E-mail addresses: ozgur.afsar@ege.edu.tr (O. Afsar), ugur.tirnakli@ege.edu.tr, utirnakli@gmail.com (U. Tirnakli). of these ‘‘particular points or regions’’, the relationship between q- Gaussians and these quantities is not completely determined and this is still an open question in the literature. To construct these relationships and to introduce some scaling laws among them can explain the reasons for the appearance of q-Gaussians. Also it is known that the standard central limit theorem (CLT) is a corner stone in statistical physics and states that, under appropriate conditions, the probability distribution of the sum of a large number of independent identically distributed (iid) random variables will be normal (or Gaussian) [12]. For a special class of strong correlations, it has been analytically shown that the standard CLT is not valid anymore due to the strongly correlated random variables and therefore the q-generalized central limit theorem (q-CLT) should be used [13]. q-Gaussian distributions, defined as, P (x) = P (0)[1 − (1 − q)β x 2 ] 1/(1−q) , (1) 0167-2789/$ – see front matter © 2014 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physd.2014.01.004