A closer look at time averages of the logistic map at the edge of chaos

TL;DR

This paper investigates the distribution of sums of logistic map iterates at the edge of chaos, confirming the convergence to q-Gaussian distributions in both the central and tail regions, and explores the scaling law involving the Feigenbaum constant.

Contribution

It provides a comprehensive numerical analysis of the entire distribution, including the central part, at the edge of chaos, supporting the applicability of nonextensive statistical mechanics.

Findings

Distribution converges to q-Gaussian in central and tail regions

Scaling law involving Feigenbaum constant delta is confirmed

Results support the use of S_q entropy 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 [see U. Tirnakli, C. Beck and C. Tsallis, 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 S_q, 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 S_q and its associated concepts.