Logistic Map--r=2

With , the logistic map becomes

(1)

The first 20 iterations of this map are illustrated above for initial values and 0.92.

The solution can be written in the form

$x_n=1/2{1-f[r^nf^(-1)(1-2x_0)]},$

(2)

with

(3)

and its inverse function (Wolfram 2002, p. 1098). Explicitly, this then gives the formula

$x_n=1/2{1-exp[2^nln(1-2x_0)]}.$

(4)

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More things to try:

MathPages. "Closed Forms for the Logistic Map." http://www.mathpages.com/home/kmath188.htm.Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, p. 1098, 2002.