Time--Evolving Statistics of Chaotic Orbits of Conservative Maps in the Context of the Central Limit Theorem

Abstract

We study chaotic orbits of conservative low--dimensional maps and present numerical results showing that the probability density functions (pdfs) of the sum of $N$ iterates in the large $N$ limit exhibit very interesting time-evolving statistics. In some cases where the chaotic layers are thin and the (positive) maximal Lyapunov exponent is small, long--lasting quasi--stationary states (QSS) are found, whose pdfs appear to converge to $q$--Gaussians associated with nonextensive statistical mechanics. More generally, however, as $N$ increases, the pdfs describe a sequence of QSS that pass from a $q$--Gaussian to an exponential shape and ultimately tend to a true Gaussian, as orbits diffuse to larger chaotic domains and the phase space dynamics becomes more uniformly ergodic.

15 pages, 14 figures, accepted for publication as a Regular Paper in the International Journal of Bifurcation and Chaos, on Jun 21, 2011

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• Canonical Ensemble in Non-extensive Statistical Mechanics
• Coupled symplectic maps as models for subdiffusive processes in disordered Hamiltonian lattices
• On the putative essential discreteness of q-generalized entropies
• A first order Tsallis theory