Power-Law Sensitivity to Initial Conditions within a Logistic-like Family of Maps: Fractality and Nonextensivity

TL;DR

This paper investigates power-law sensitivity to initial conditions in logistic-like maps at critical points, revealing a relationship between the entropic index q and the fractal dimension of the attractor, bridging nonextensive statistics and fractal geometry.

Contribution

It establishes a connection between the entropic index q and the fractal dimension d_f in logistic-like maps, extending the understanding of sensitivity at criticality within nonextensive statistics.

Findings

q decreases monotonically from 1 to -infinity as d_f decreases from 1 to 0

Power-law sensitivity characterized by a generalized deviation law

Relation between nonextensive parameter q and fractal dimension d_f

Abstract

Power-law sensitivity to initial conditions, characterizing the behaviour of dynamical systems at their critical points (where the standard Liapunov exponent vanishes), is studied in connection with the family of nonlinear 1D logistic-like maps $x_{t+1} = 1 - a | x_t |^z, (z > 1; 0 < a \le 2; t=0,1,2,...)$ The main ingredient of our approach is the generalized deviation law $\lim_{\Delta x(0) -> 0} \Delta x(t) / \Delta x(0)} = [1+(1-q)\lambda_q t]^{1/(1-q)}$ (equal to $e^{\lambda_1 t}$ for q=1, and proportional, for large t, to $t^{1/(1-q)}$ for $q \ne 1; q \in R$ is the entropic index appearing in the recently introduced nonextensive generalized statistics). The relation between the parameter q and the fractal dimension d_f of the onset-to-chaos attractor is revealed: q appears to monotonically decrease from 1 (Boltzmann-Gibbs, extensive, limit) to -infinity when d_f varies from 1 (nonfractal, ergodic-like, limit) to zero.