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 valid only 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.

As well known, Boltzmann-Gibbs statistics is the correct way of thermostatistically approaching ergodic systems. On the other hand, nontrivial ergodicity breakdown and strong correlations typically drag the system into out-of-equilibrium states where Boltzmann-Gibbs statistics fails. For a wide class of such systems, it has been shown in recent years that the correct approach is to use Tsallis statistics instead. Here we show how the dynamics of the paradigmatic conservative (area-preserving) stan-dard map exhibits, in an exceptionally clear manner, the crossing from one statistics to the other. Our results unambiguously illustrate the domains of validity of both Boltzmann-Gibbs and Tsallis statistical distributions. Since various important physical systems from particle confinement in magnetic traps to autoionization of molecular Rydberg states, through particle dynamics in accelerators and comet dynamics, can be reduced to the standard map, our results are expected to enlighten and enable an improved interpretation of diverse experimental and observational results.

For a family of logistic-like maps, we investigate the rate of convergence to the critical attractor when an ensemble of initial conditions is uniformly spread over the entire phase space. We found that the phase space volume occupied by the ensemble W (t) depicts a power-law decay with log-periodic oscillations reflecting the multifractal character of the critical attractor. We explore the parametric dependence of the power-law exponent and the amplitude of the log-periodic oscillations with the attractor's fractal dimension governed by the inflexion of the map near its extremal point. Further, we investigate the temporal evolution of W (t) for the circle map whose critical attractor is dense. In this case, we found W (t) to exhibit a rich pattern with a slow logarithmic decay of the lower bounds. These results are discussed in the context of non-extensive Tsallis entropies.

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 75, 040106(R) (2007)] to be numerically consistent with a q-Gaussian, the distribution which, under appropriate constraints, maximizes the nonadditive entropy Sq, 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 δ. 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. One of the cornerstones of statistical mechanics and of probability theory is the Central Limit Theorem (CLT). It states that the sum of N independent identically distributed random variables, after appropriate centering and rescaling, approaches a Gaussian distribution as N → ∞. In general, this concept lies at the very heart of the fact that many stochastic processes in nature which consist of a sum of many independent or nearly independent variables converge to a Gaussian process [1,2]. On the other hand, there are also many other occasions in nature for which the limit distribution is not a Gaussian. The common ingredient for such systems is the existence of strong correlations between the random variables, which prevent the limit distribution of the system to end up being a Gaussian. Recently, for certain classes of strong correlations of this kind, it has been proved that the distribution of the rescaled sum approaches a q-Gaussian, which constitutes a q-generalization of the standard CLT [3,4,5,6]. This represents a progress since the q-Gaussians are the distributions that optimize the nonadditive entropy S q (defined to be S q ≡ (1 − i p q i ) / (q − 1)), on which nonextensive statistical mechanics is based [7,8]. A q-generalized CLT was expected for several years since the role of q-Gaussians in nonextensive statistical mechanics is pretty much the same as that of Gaussians in Boltzmann-Gibbs statistical mechanics. Therefore it is not surprising at all to see q-Gaussians replace the usual Gaussian distributions for those systems whose agents exhibit certain types of strong correlations.Immediately after these achievements, an increasing interest developed for checking these ideas and findings in real and model systems whose dynamical properties make them appropriate candidates to be analyzed along these l...