Nonextensivity at the edge of chaos of a new universality class of one-dimensional unimodal dissipative maps

We introduce a new universality class of one-dimensional unimodal dissipative maps. The new family, from now on referred to as the (z 1 ​ ,z 2 ​ )-logarithmic map, corresponds to a generalization of the z-logistic map. The Feigenbaum-like constants of these maps are determined. It has been recently shown that the probability density of sums of iterates at the edge of chaos of the z-logistic map is numerically consistent with a q-Gaussian, the distribution which, under appropriate constraints, optimizes the nonadditive entropy S q ​ . We focus here on the presently generalized maps to check whether they constitute a new universality class with regard to q-Gaussian attractor distributions. We also study the generalized q-entropy production per unit time on the new unimodal dissipative maps, both for strong and weak chaotic cases. The q-sensitivity indices are obtained as well. Our results are, like those for the z-logistic maps, numerically compatible with the q-generalization of a Pesin-like identity for ensemble averages.

2009