Closer look at time averages of the logistic map at the edge of chaos


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Tirnakli U. , TSALLIS C., Beck C.

PHYSICAL REVIEW E, vol.79, no.5, 2009 (Journal Indexed in SCI) identifier identifier identifier

  • Publication Type: Article / Article
  • Volume: 79 Issue: 5
  • Publication Date: 2009
  • Doi Number: 10.1103/physreve.79.056209
  • Title of Journal : PHYSICAL REVIEW E

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 S(q), 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 S(q) and its associated concepts.