Closer look at time averages of the logistic map at the edge of chaos
Makaleler > Tam Makale · 2009
Yazarlar: Uğur TIRNAKLI, Constantino TSALLIS, Christian Beck
Physical Review E, SCI Expanded İndekslerine Giren Dergi, Uluslararası, Wos, 2009, cilt 79, ss 056209-056209
DOI: 10.1103/physreve.79.056209
Özet
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.