%0 Book Section %1 statphys23_0029 %A Tirnakli, U. %A Beck, C. %A Tsallis, C. %B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics %C Genova, Italy %D 2007 %E Pietronero, Luciano %E Loreto, Vittorio %E Zapperi, Stefano %K chaos chaotic classical low-dimensional mechanics numerical simulations statistical statphys23 systems topic-1 %T Central limit behavior of deterministic dynamical systems %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=29 %X We investigate the probability density of rescaled sums of iterates of deterministic dynamical systems, a problem relevant for many complex physical systems consisting of dependent random variables. A Central Limit Theorem (CLT) is only valid if the dynamical system under consideration is sufficiently mixing. For the fully developed logistic map and a cubic map we analytically calculate the leading-order corrections to the CLT if only a finite number of iterates is added and rescaled, and find excellent agreement with numerical experiments. At the critical point of period doubling accumulation, a CLT is not valid anymore due to strong temporal correlations between the iterates. Nevertheless, we provide numerical evidence that in this case the probability density converges to a $q$-Gaussian, thus leading to a power-law generalization of the CLT. The above behavior is universal and independent of the order of the maximum of the map considered, i.e., relevant for large classes of critical dynamical systems.\\ References \\ U.Tirnakli, C.Beck and C.Tsallis, cond-mat/0701622, Phys. Rev. E / Rapid Comm., (2007), in press.

@incollection{statphys23_0029, abstract = {We investigate the probability density of rescaled sums of iterates of deterministic dynamical systems, a problem relevant for many complex physical systems consisting of dependent random variables. A Central Limit Theorem (CLT) is only valid if the dynamical system under consideration is sufficiently mixing. For the fully developed logistic map and a cubic map we analytically calculate the leading-order corrections to the CLT if only a finite number of iterates is added and rescaled, and find excellent agreement with numerical experiments. At the critical point of period doubling accumulation, a CLT is not valid anymore due to strong temporal correlations between the iterates. Nevertheless, we provide numerical evidence that in this case the probability density converges to a $q$-Gaussian, thus leading to a power-law generalization of the CLT. The above behavior is universal and independent of the order of the maximum of the map considered, i.e., relevant for large classes of critical dynamical systems.\\ References \\ U.Tirnakli, C.Beck and C.Tsallis, cond-mat/0701622, Phys. Rev. E / Rapid Comm., (2007), in press.}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Tirnakli, U. and Beck, C. and Tsallis, C.}, biburl = {https://www.bibsonomy.org/bibtex/263b8a868078bafe83f59bf6922a1685a/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {9083847b1afb9196e14c453054119d31}, intrahash = {63b8a868078bafe83f59bf6922a1685a}, keywords = {chaos chaotic classical low-dimensional mechanics numerical simulations statistical statphys23 systems topic-1}, month = {9-13 July}, timestamp = {2007-06-20T10:16:10.000+0200}, title = {Central limit behavior of deterministic dynamical systems}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=29}, year = 2007 }