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

(2)

with

(3)

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

(4)

REFERENCES:

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.