Logistic Map--r=-2

With , the logistic map becomes

(1)

The first 50 iterations of this map are illustrated above for initial values  and 0.4.

The solution can be written in the form

(2)

with

			(3)
			(4)

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

(5)

 has the Maclaurin series

			(6)
			(7)

(OEIS A059944).

REFERENCES:

MathPages. "Closed Forms for the Logistic Map." http://www.mathpages.com/home/kmath188.htm.

Sloane, N. J. A. Sequence A059944 in "The On-Line Encyclopedia of Integer Sequences."

Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, p. 1098, 2002.