Logistic Map--r=4

With , the logistic map becomes

(1)

which is equivalent to the tent map with . The first 50 iterations of this map are illustrated above for initial values  and 0.71.

The solution can be written in the form

(2)

with

(3)

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

			(4)
			(5)
			(6)

To investigate the equation's properties, let

(7)

(8)

(9)

so

(10)

Manipulating (7) gives

			(11)
			(12)

so

(13)

(14)

But . Taking , then  and

(15)

For ,  and

(16)

Combining gives

(17)

which can be written

(18)

which is just the tent map with , whose natural invariant in  is

(19)

Transforming back to  therefore gives

			(20)
			(21)
			(22)

This can also be derived from

(23)

where  is the delta function.

REFERENCES:

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

 Jaffe, S. "The Logistic Map: Computable Chaos." http://library.wolfram.com/infocenter/MathSource/579/.

Whittaker, J. V. "An Analytical Description of Some Simple Cases of Chaotic Behavior." Amer. Math. Monthly 98, 489-504, 1991.

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