Properties of the class of power-logistic maps

Abstract

In a study of the class of power-logistic maps, each of which consists of a power-law branch 1 -r|xn|z for negative values of Xn and a quadratic (logistic) branch 1- rxn 2 for positive values of xn with parameter r ε (0,2] and exponent z ε (0,2], we found the following: (i) In the chaotic region, there are stable cycles whose periods can be regarded to form an arithmetic progression known as pattern A (PA). (ii) As z decreases, PA is more prominent; nevertheless, it still exists in the logistic map. (iii) The first term of PA is a function of z: as z decreases, it either stays constant or increases by 2. (iv) As z decreases, a given PA term begins to appear at a smaller r value, (v) When z is sufficiently large, the range of a PA term increases as z decreases. (vi) Between two consecutive PA terms, there are structures such as period-doubled cycles of the PA terms, other stable cycles, and a chaotic subregion. (vii) As z decreases, the chaotic subregion between any two consecutive PA terms shrinks, which may result in a loss of fine structures.