Nonextensivity at the edge of chaos of a new universality class of one-dimensional unimodal dissipative maps

Abstract

We introduce a new universality class of one-dimensional unimodal dissipative maps. The new family, from now on referred to as the ($z_1,z_2$)-{\it logarithmic map}, corresponds to a generalization of the $z$-logistic map. The Feigenbaum-like constants of these maps are determined. It has been recently shown that the probability density of sums of iterates at the edge of chaos of the $z$-logistic map is numerically consistent with a $q$-Gaussian, the distribution which, under appropriate constraints, optimizes the nonadditive entropy $S_q$. We focus here on the presently generalized maps to check whether they constitute a new universality class with regard to $q$-Gaussian attractor distributions. We also study the generalized $q$-entropy production per unit time on the new unimodal dissipative maps, both for strong and weak chaotic cases. The $q$-sensitivity indices are obtained as well. Our results are, like those for the $z$-logistic maps, numerically compatible with the $q$-generalization of a Pesin-like identity for ensemble averages.

arXiv comment17 pages, 10 figures. To appear in European Physical Journal B

References in corpus (9)

• Superdiffusion and non-Gaussian statistics in a driven-dissipative 2D dusty plasma
• The rate of entropy increase at the edge of chaos
• Central limit behavior of deterministic dynamical systems
• A closer look at time averages of the logistic map at the edge of chaos
• Ensemble averages and nonextensivity at the edge of chaos of one-dimensional maps
• Deterministic Brownian Motion: The Effects of Perturbing a Dynamical System by a Chaotic Semi-Dynamical System
• Dynamics towards the Feigenbaum attractor
• q-Deformed Statistical-Mechanical Property in the Dynamics of Trajectories en route to the Feigenbaum Attractor
• Critical attractors and q-statistics

Cited by in corpus (18)

• Black hole thermodynamical entropy
• Nonadditive entropy: the concept and its use
• Reply to Comment on "Towards a large deviation theory for strongly correlated systems"
• Central Limit Behavior in the Kuramoto model at the 'Edge of Chaos'
• Probability densities for the sums of iterates of the sine-circle map in the vicinity of the quasi-periodic edge of chaos
• Time--Evolving Statistics of Chaotic Orbits of Conservative Maps in the Context of the Central Limit Theorem
• Generalized Huberman-Rudnick scaling law and robustness of $q$-Gaussian probability distributions
• Thermodynamics is more powerful than the role to it reserved by Boltzmann-Gibbs statistical mechanics
• 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
• Analysis of Tsallis' classical partition function's poles
• Asymptotic expansions for the escape rate of stochastically perturbed unimodal maps
• Self-similar motion for modeling anomalous diffusion and nonextensive statistical distributions
• Rescuing the MaxEnt treatment for $q$-generalized entropies
• Verlinde's emergent gravity in an $\mathbf{n}-$dimensional, non-additive Tsallis' scenario
• Hidden correlations entailed by q-non additivity render the q-monoatomic gas highly non trivial
• q-Path entropy phenomenology for phase-space curves
• Physical peculiarities of divergences emerging in q-deformed statistics