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A closer look at time averages of the logistic map at the edge of chaos | | Ugur Tirnakli
; Constantino Tsallis
; Christian Beck
; | | Date: |
8 Feb 2008 | | 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 {f 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. | | Source: | arXiv, 0802.1138 | | Services: | Forum | Review | PDF | Favorites | |
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