Logistic Map--r=4 -- from Wolfram MathWorld

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

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

(2)

with

(3)

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

To investigate the equation's properties, let

(7)

(8)

(9)

(10)

Manipulating (7) gives

		$41/2[1-cos(piy_n)]{1-1/2[1-1/2(1-cos(piy_n))]}$	(11)
			(12)

(13)

(14)

But . Taking , then and

(15)

For , and

(16)

Combining gives

$y_(n+1)={2y_n for y_n in [0,1/2]; 2-2y_n for y_n in [1/2,1],$

(17)

which can be written

(18)

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

(19)

Transforming back to therefore gives

This can also be derived from

(23)

References

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.

Logistic Map--r=4