Sensitivity function and entropy increase rates for z-logistic map family at the edge of chaos

Abstract

It is well known that, for chaotic systems, the production of relevant entropy (Boltzmann-Gibbs) is always linear and the system has strong (exponential) sensitivity to initial conditions. In recent years, various numerical results indicate that basically the same type of behavior emerges at the edge of chaos if a specific generalization of the entropy and the exponential are used. In this work, we contribute to this scenario by numerically analysing some generalized nonextensive entropies and their related exponential definitions using z-logistic map family. We also corroborate our findings by testing them at accumulation points of different cycles.

Vanishing largest Lyapunov exponent and Tsallis entropy

“…This is what we wanted to show. We see that (35) is the analogue of (23) for (X, d). Naturally, the argument leading to (35) is already applicable to the case of a Riemannian manifold (M, g) of negative sectional curvature, since in this case (M, g) is a CAT(k), k < 0 space.…”

“…We see that (35) is the analogue of (23) for (X, d). Naturally, the argument leading to (35) is already applicable to the case of a Riemannian manifold (M, g) of negative sectional curvature, since in this case (M, g) is a CAT(k), k < 0 space. So, the conclusions drawn in the Riemannian case, about the effective metric behavior of systems described by the Tsallis entropy, can be extended unaltered to the case of any number of interacting systems described by different values of q.…”

“…The conclusion that we reach from (35), is that the number of points with a uniform distance upper bound ǫ between two consecutive ones, making up the path between y and z outside B r (x) increases exponentially with r in the CAT(k), k < 0 space (X, d). This is what we wanted to show.…”

Theoretical Foundations and Mathematical Formalism of the Power-Law Tailed Statistical Distributions

“…the cosmic rays [3], relativistic [37] and classical [38] plasmas in presence of external electromagnetic fields, the relaxation in relativistic plasmas under wave-particle interactions [39,40], anomalous diffusion [41,42], nonlinear kinetics [43][44][45], kinetics of interacting atoms and photons [46], particle kinetics in the presence of temperature gradients [47,48], particle systems in external conservative force fields [49], stellar distributions in astrophysics [50][51][52][53], quark-gluon plasma formation [54], quantum hadrodynamics models [55], the fracture propagation [56], etc. Other applications concern dynamical systems at the edge of chaos [57][58][59], fractal systems [60], field theories [61], the random matrix theory [62][63][64], the error theory [65], the game theory [66], the theory of complex networks [67], the information theory [68], etc. Also applications to economic systems have been considered e.g.…”

Introduction to Nonextensive Statistical Mechanics

“…The properties that have been exhibited here for the sensitivity to the initial conditions and the entropy production have also been checked [143,144] for other entropies directly related to S q . The scheme remains the same, excepting for the slope K q ent , which does depend on the particular entropy.…”