Relationships and scaling laws among correlation, fractality, Lyapunov divergence and [formula omitted]-Gaussian distributions

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 (Dcorr=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∝lτ is evident with the critical exponent τ=0.23±0.01. Similarly, as chaos threshold is approached, the quantity l scales as l∝(a−ac)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∝λμ) and the distance to the critical box counting fractal dimension (1/g∝(D−Dc)μ) with the same exponent μ≅0.43. Finally, we numerically verify that these three quantities (ξ, λ, D−Dc) scale with the distance to the critical control parameter of the map (i.e., a−ac) 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.