Logistic Map--r=4
المؤلف:
MathPages.
المصدر:
"Closed Forms for the Logistic Map." http://www.mathpages.com/home/kmath188.htm.
الجزء والصفحة:
...
22-12-2021
2687
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<span style=]() {1-f[r^nf^(-1)(1-2x_0)]}, " src="https://mathworld.wolfram.com/images/equations/LogisticMapR=4/NumberedEquation2.gif" style="height:23px; width:193px" /> |
(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
![x=sin^2(1/2piy)=1/2[1-cos(piy)]](https://mathworld.wolfram.com/images/equations/LogisticMapR=4/NumberedEquation4.gif) |
(7)
|
 |
(8)
|
 |
(9)
|
so
 |
(10)
|
Manipulating (7) gives
so
 |
(13)
|
 |
(14)
|
But ![y in [0,1]](https://mathworld.wolfram.com/images/equations/LogisticMapR=4/Inline20.gif)
. Taking ![y_n in [0,1/2]](https://mathworld.wolfram.com/images/equations/LogisticMapR=4/Inline21.gif)
, then 
and
 |
(15)
|
For ![y in [1/2,1]](https://mathworld.wolfram.com/images/equations/LogisticMapR=4/Inline23.gif)
, 
and
 |
(16)
|
Combining gives
![y_(n+1)=<span style=]() {2y_n for y_n in [0,1/2]; 2-2y_n for y_n in [1/2,1], " src="https://mathworld.wolfram.com/images/equations/LogisticMapR=4/NumberedEquation12.gif" style="height:56px; width:200px" /> |
(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)
|
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.
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