Analytical demonstration of the q-deformed central limit property of the Feigenbaum attractor
Abstract Book of the XXIII IUPAP International Conference on Statistical Physics, Genova, Italy, (9-13 July 2007)Zusammenfassung
We examine the development of the limiting form of the probability density of renormalized sums of positions associated to the Feigenbaum attractor. We prove analytically that this limiting form conforms to a q-deformed modification of the gaussian distribution of the ordinary central limit theorem (CLT). The procedure is based on the determination of the increase of numbers of diameters (or distances of neighboring positions) of asymptotically-equal lengths and of the concomitant power-law decrease of these lengths as the Feigenbaum cascade develops. The tails of the resultant distribution have a precise value of the q index, q=1+ln2/ln $\alpha$ 1.755, where $\alpha$= 2.5029
is Feigenbaum's universal constant. Our findings support recent numerical studies 1 that point at a specific modification of the CLT displayed by chaotic attractors, akin to independent random variables systems, to the case of critical attractors, where persistent memory of positions plays a role similar to that of correlated random variables in a stochastic system.
1) U. Tirnakli, C. Beck, C. Tsallis, Central limit behavior of deterministic dynamical systems, Phys. Rev. E (in press), and cond-mat/0701622