title: Closer look at time averages of the logistic map at the edge of chaos.
abstract: The probability distribution of sums of iterates of the logistic map at the edge of chaos has been recently shown [U. Tirnakli, Phys. Rev. E 75, 040106(R) (2007)] to be numerically consistent with a q-Gaussian, the distribution which-under appropriate constraints-maximizes the nonadditive entropy Sq, which is the basis of nonextensive statistical mechanics. This analysis was based on a study of the tails of the distribution. We now check the entire distribution, in particular, its central part. This is important in view of a recent q generalization of the central limit theorem, which states that for certain classes of strongly correlated random variables the rescaled sum approaches a q-Gaussian limit distribution. We numerically investigate for the logistic map with a parameter in a small vicinity of the critical point under which conditions there is convergence to a q-Gaussian both in the central region and in the tail region and find a scaling law involving the Feigenbaum constant delta. Our results are consistent with a large number of already available analytical and numerical evidences that the edge of chaos is well described in terms of the entropy Sq and its associated concepts.
authors.affiliation: Department of Physics, Faculty of Science, Ege University, 35100 Izmir, Turkey. ugur.tirnakli@ege.edu.tr
authors.forename: Ugur
authors.initials: U
authors.lastname: Tirnakli
authors.forename: Constantino
authors.initials: C
authors.lastname: Tsallis
authors.forename: Christian
authors.initials: C
authors.lastname: Beck