Generalising the logistic map through the $q$-product

TL;DR

This paper generalizes the logistic map using the $q$-product, exploring its dynamics at the edge of chaos and connecting it with nonextensive statistical mechanics.

Contribution

It introduces a $q$-product based generalization of the logistic map and analyzes its behavior at the critical point, linking it to nonextensive statistical mechanics.

Findings

Bifurcation diagrams reveal complex dynamics for different $q_{map}$ values.

Sensitivity to initial conditions varies with $q_{map}$, indicating different chaos regimes.

Entropy growth rates align with nonextensive statistical mechanics predictions.

Abstract

We investigate a generalisation of the logistic map as $ x_{n+1}=1-ax_{n}\otimes_{q_{map}} x_{n}$ ($-1 \le x_{n} \le 1$, $0<a\le2$) where $\otimes_q$ stands for a generalisation of the ordinary product, known as $q$-product [Borges, E.P. Physica A {\bf 340}, 95 (2004)]. The usual product, and consequently the usual logistic map, is recovered in the limit $q\to 1$, The tent map is also a particular case for $q_{map}\to\infty$. The generalisation of this (and others) algebraic operator has been widely used within nonextensive statistical mechanics context (see C. Tsallis, {\em Introduction to Nonextensive Statistical Mechanics}, Springer, NY, 2009). We focus the analysis for $q_{map}>1$ at the edge of chaos, particularly at the first critical point $a_c$, that depends on the value of $q_{map}$. Bifurcation diagrams, sensitivity to initial conditions, fractal dimension and rate of entropy growth are evaluated at $a_c(q_{map})$, and connections with nonextensive statistical mechanics are explored.