On logistic-like iterative maps
Logistic-like first order iterative maps, defined here as Xn+1 = r Xnλ(1-Xn)μ, are examined. The parameters r, λ and μ are positive real numbers, while the variable x and its map range from 0 to 1, the latter yielding the upper value of r for which full chaos is obtained. Depending on the values of λ and μ, the resulting x's can have a totally different behavior from those of the logistic map, given by λ = μ =1. The focus here is on fixed points since their existence, for given values of λ and μ, is necessary for obtaining chaotic x's. The purpose of the paper is four-fold: first, to define regions of existence for the fixed point(s) in terms of the parameters r, λ and μ; second, to determine the nature of the fixed points, whether they are attractors (stable), repellors (unstable) or super-stable, according to the values of the parameters; third, to define those maps for which the fixed points can be written in explicit algebraic form; and fourth, for iterative nearby maps, to obtain their fixed points in an approximate algebraic form in terms of the exact fixed points. The approximation is based on Newton's method, one step from the nearest iterative map whose fixed points can be obtained exactly, in explicit form. The validity of the fixed point approximation depends on the stability of the fixed points and is, subsequently, established in respect of well defined surfaces of the parameters r, λ and μ.
