Generalising the logistic map through the $q$-product

Abstract

We investigate a generalisation of the logistic map as $x_{n+1}=1-a{x}_n{\otimes }_{q_{map}}{x}_n$ ($-1\le x_n\le 1$, $0<a\le 2$) 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.

12 pages, 23 figures, Dynamics Days South America. To be published in Journal of Physics: Conference Series (JPCS - IOP)

References in corpus (5)

• A possible deformed algebra and calculus inspired in nonextensive thermostatistics
• Central limit behavior of deterministic dynamical systems
• A closer look at time averages of the logistic map at the edge of chaos
• Two-dimensional dissipative maps at chaos threshold: Sensitivity to initial conditions and relaxation dynamics
• Mixing and relaxation dynamics of the Henon map at the edge of chaos

Cited by in corpus (1)

• Distinguishability notion based on Wootters statistical distance: application to discrete maps