Limit behaviour and scaling relations of two kinds of noisy logistic map in the vicinity of chaos threshold and their robustness

In this paper we numerically investigate the distribution of the sums of the iterates of the logistic map and the relationships among the important properties of the nonlinear dynamics in the vicinity of the chaos threshold by adding two kinds of contributions with different densities. The first one is the well-known white noise, whereas the second is a newly defined one, named as quartic term, which makes contributions from the own structure of the map. As the chaos threshold is approached, the iterates of the standard logistic map (i.e. noise-free) have strong correlations and the standard Central Limit Theorem is not valid anymore. In a recent work (Tirnakli, 2009), it has been shown that the limit distribution seems to converge to q-Gaussian distribution, which maximizes the nonadditive entropy Sq≡(1−∑ipiq)/(q−1) under appropriate conditions. In this work, we investigate the effect of these contributions (i.e. white noise and quartic term) on the limit distribution and on the range of the obtained q-Gaussian distribution. As a result of these findings, under the existence of white noise and also the quartic term, we analyse the validity of the scaling relations among correlation, fractality, the Lyapunov divergence and q-Gaussian distributions, which have recently been observed in (Afsar, 2014). The results obtained here strengthen the argument that the central limit behaviour is given by a q-Gaussian as the chaos threshold is approached and indicate that the scaling relations, obtained for the standard logistic map, among the range of the q-Gaussian, the correlation dimension, the correlation length, the Lyapunov exponent, fractality and the distance from the chaos threshold are robust under the existence of white noise and the quartic term.