The Feigenbaum diagram

The Feigenbaum diagram plots the stable orbits of the logistic iteration map [math]f(x)=px(1-x)[/math], where [math]0\leq p\leq 4[/math] and [math]0\leq x\leq 1[/math]:[br]Given any initial value [math]x_0[/math], the iterations of the map give the sequence [math]\{x_n|n\in\mathbb{N}\}[/math], where [math]x_n=f(x_{n-1})[/math] for [math]n>0[/math].[br]After an initial transient phase, for sufficiently large [math]n[/math] and for suitable values ​​of [math]p[/math] the orbit stabilises: the applet shows for any value of [math]p[/math] (on the abscissa axis) the values of [math]x_{101}, x_{102}, x_{103},\cdots, x_{115}, x_{116}[/math] (on the ordinate axis).
The Feigenbaum diagram

Information: The Feigenbaum diagram