In a study of the class of power-logistic maps, each of which consists of a power-law branch $1-r{\left|x_n\right|}^z$ for negative values of $x_n$ and a quadratic (logistic) branch $1-r{x}_n^2$ for positive values of $x_n$ with parameter $r\in (0, 2]$ and exponent $z\in (0, 2]$, 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.

• Received 13 May 1996

DOI:https://doi.org/10.1103/PhysRevE.54.5985