TITLE: Show that the familiar logistic map $x_{n+1} = sx_n(1 - x_n)$, can be recoded into the form $x_{n+1} = x_n^2 + c$.
QUESTION [6 upvotes]: What change of variables would trtansform the logistic equation into the Mandelbrot equation $z_{n+1}=z_n^2+c$?

REPLY [2 votes]: Using the change of variable $z_n=s\cdot(\frac12-x_n)$ yields $z_{n+1}=z_n^2+c$ with $c=\frac12s-\frac14s^2$.