Relationships and scaling laws among correlation, fractality, Lyapunov divergence and q-Gaussian distributions

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 proportional to l(tau) is evident with the critical exponent tau = 0.23 +/- 0.01. Similarly, as chaos threshold is approached, the quantity l scales as l proportional to (a - a(c))(gamma), where the exponent is gamma = 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 proportional to xi(-mu)), Lyapunov divergence (1/g proportional to lambda(mu)) and the distance to the critical box counting fractal dimension (1/g proportional to (D - D-c)(mu)) with the same exponent mu congruent to 0.43. Finally, we numerically verify that these three quantities (xi, lambda, 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 v = 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. (C) 2014 Elsevier B.V. All rights reserved.