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Date: 12-10-2021
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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) |
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(5) |
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(6) |
To investigate the equation's properties, let
(7) |
(8) |
(9) |
so
(10) |
Manipulating (7) gives
(11) |
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(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) |
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(21) |
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(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.
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