Logistic Map--r=-2
المؤلف:
MathPages
المصدر:
"Closed Forms for the Logistic Map." http://www.mathpages.com/home/kmath188.htm.
الجزء والصفحة:
...
22-12-2021
2628
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
![x_n=1/2<span style=]() {1-f[r^nf^(-1)(1-2x_0)]}, " src="https://mathworld.wolfram.com/images/equations/LogisticMapR=-2/NumberedEquation2.gif" style="height:23px; width:193px" /> |
(2)
|
with
and 
its inverse function (Wolfram 2002, p. 1098). Explicitly, this then gives the formula
![x_n=1/2-cos<span style=]() {1/3[pi-(-2)^n(pi-3cos^(-1)(1/2-x_0))]}. " src="https://mathworld.wolfram.com/images/equations/LogisticMapR=-2/NumberedEquation3.gif" style="height:23px; width:288px" /> |
(5)
|
has the Maclaurin series
(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.
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