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Logistic Map--r=-2

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LogisticEquation-2

The logistic map with r=-2 is the recurrence equation

x_(n+1)=-2x_n(1-x_n).
(1)

The first 50 iterations of this map are illustrated above for initial values x_0=0.31 and 0.4.

The solution can be written in the form

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

with

f(x)=cos(x/(sqrt(3)))+sqrt(3)sin(x/(sqrt(3)))
(3)
=2cos(1/3(pi-sqrt(3)x))
(4)

and f^(-1)(y)=[pi-3cos^(-1)(y/2)]/sqrt(3) is a branch of its inverse function (Wolfram 2002, p. 1098). Explicitly, this then gives the formula

x_n=1/2-cos{1/3[pi-(-2)^n(pi-3cos^(-1)(1/2-x_0))]}.
(5)

Here cos^(-1) is the inverse cosine. The function f(x) has the Maclaurin series

f(x)=sum_(n=0)^(infty)(3^(-n/2))/(n!)[cos(1/2npi)+sqrt(3)sin(1/2pin)]x^n
(6)
=1+x-1/6x^2-1/(18)x^3+1/(216)x^4+1/(1080)x^5-1/(19440)x^6+...
(7)

(OEIS A059944).

See also

Logistic Map, Logistic Map--r=2, Logistic Map--r=4

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References

MathPages. "Closed Forms for the Logistic Map." https://www.mathpages.com/home/kmath011/kmath011.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.

Referenced on Wolfram|Alpha

Logistic Map--r=-2

Cite this as:

Weisstein, Eric W. "Logistic Map--r=-2." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LogisticMapR=-2.html

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