Central limit behavior of deterministic dynamical systems

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.

6 pages, 5 figures

References in corpus (4)

• Deterministic Brownian Motion: The Effects of Perturbing a Dynamical System by a Chaotic Semi-Dynamical System
• Criticality in non-linear one-dimensional maps: RG universal map and non-extensive entropy
• Non-extensive Behavior of a Stock Market Index at Microscopic Time Scales
• Occupancy of phase space, extensivity of Sq, and q-generalized central limit theorem

Cited by in corpus (48)

• Nonadditive entropy: the concept and its use
• The standard map: From Boltzmann-Gibbs statistics to Tsallis statistics
• A closer look at time averages of the logistic map at the edge of chaos
• Nonergodicity and Central Limit Behavior for Long-range Hamiltonians
• Limit distributions of scale-invariant probabilistic models of correlated random variables with the q-Gaussian as an explicit example
• A closer look at the indications of q-generalized Central Limit Theorem behavior in quasi-stationary states of the HMF model
• Strictly and asymptotically scale-invariant probabilistic models of $N$ correlated binary random variables having {\em q}--Gaussians as $N\to \infty$ limiting distributions
• Thermostatistics of deformed bosons and fermions
• Intermediate statistics as a consequence of deformed algebra
• Nonadditive entropy and nonextensive statistical mechanics - Some central concepts and recent applications
• Reply to Comment on "Towards a large deviation theory for strongly correlated systems"
• q-Generalization of the inverse Fourier transform
• Central Limit Behavior in the Kuramoto model at the 'Edge of Chaos'
• Nonextensivity at the edge of chaos of a new universality class of one-dimensional unimodal dissipative maps
• Probability densities for the sums of iterates of the sine-circle map in the vicinity of the quasi-periodic edge of chaos
• Ergodicity and Central Limit Theorem in Systems with Long-Range Interactions
• Time--Evolving Statistics of Chaotic Orbits of Conservative Maps in the Context of the Central Limit Theorem
• Comment on "Central limit behavior in deterministic dynamical systems"
• Generalized Huberman-Rudnick scaling law and robustness of $q$-Gaussian probability distributions
• On the non-Boltzmannian nature of quasi-stationary states in long-range interacting systems
• The maximization of Tsallis entropy with complete deformed functions and the problem of constraints
• Statistical characterization of the standard map
• Nonextensivity in the solar magnetic activity during the increasing phase of solar Cycle 23
• Inversion of Tsallis' q-Fourier Transform and the complex-plane generalization
• On the role of ergodicity and mixing in the central limit theorem for Casati-Prosen triangle map variables
• Inter-occurrence times and universal laws in finance, earthquakes and genomes
• Renormalization group structure for sums of variables generated by incipiently chaotic maps
• Nonextensive triplet in geological faults system
• Reflections on the q-Fourier transform and the q-Gaussian function
• Long memory constitutes a unified mesoscopic mechanism consistent with nonextensive statistical mechanics
• On "Ergodicity and Central Limit Theorem in Systems with Long-Range Interactions" by Figueiredo et al
• On the robustness of the $q$-Gaussian family
• Central limit theorem for a class of globally correlated random variables
• On the relevance of q-distribution functions: The return time distribution of restricted random walker
• Sums of variables at the onset of chaos
• A Shannon-Tsallis transformation
• The Tsallis-Laplace Transform
• Dynamical robustness of discrete conservative systems: Harper and generalized standard maps
• Self-similar motion for modeling anomalous diffusion and nonextensive statistical distributions
• Scaling of distributions of sums of positions for chaotic dynamics at band-splitting points
• Generalising the logistic map through the $q$-product
• Nilpotence in Physics: the case of Tsallis entropy
• The available force in long-range interaction complex systems and its statistical physical properties
• Information geometry and synchronization phase transition in Kuramoto model
• Stationary distributions of sums of marginally chaotic variables as renormalization group fixed points
• Sums of variables at the onset of chaos, replenished
• An alternative to the coupling of Berkes-Liu-Wu for strong approximations
• T. Dauxois' "Non-Gaussian distributions under scrutiny" under scrutiny