Relationships and scaling laws among correlation, fractality, Lyapunov divergence and -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 c o r r = 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.