Logistic Equation r=4

With , the Logistic Equation becomes

$\begin{displaymath} x_{n+1}=4x_n(1-x_n). \end{displaymath}$

(1)

Now let

$\begin{displaymath} x\equiv \sin^2({\textstyle{1\over 2}}\pi y) = {\textstyle{1\over 2}}[1-\cos(\pi y)] \end{displaymath}$

(2)

$\begin{displaymath} \sqrt{x}=\sin({\textstyle{1\over 2}}\pi y) \end{displaymath}$

(3)

$\begin{displaymath} y={2\over\pi}\sin^{-1}(\sqrt{x}\,) \end{displaymath}$

(4)

$\begin{displaymath} {dy\over dx}={2\over\pi} {1\over\sqrt{1-x}} {\textstyle{1\over 2}}x^{-1/2} = {1\over\pi\sqrt{x(1-x)}}. \end{displaymath}$

(5)

Manipulating (2) gives

$\displaystyle \sin^2({\textstyle{1\over 2}}\pi y_{n+1})$	$\textstyle =$	$\displaystyle 4 {\textstyle{1\over 2}}[1-\cos(\pi y_n)]\{1-{\textstyle{1\over 2}}[1-{\textstyle{1\over 2}}(1-\cos(\pi y_n)]\}$
	$\textstyle =$	$\displaystyle 2[1-\cos(\pi y = 1-\cos^2(\pi y_n)\sin^2(\pi y_n),$	(6)

$\begin{displaymath} {\textstyle{1\over 2}}\pi y_{n+1} =\pm y_n +s\pi \end{displaymath}$

(7)

$\begin{displaymath} y_{n+1}=\pm 2 y_n+{\textstyle{1\over 2}}s. \end{displaymath}$

(8)

But $y\in [0,1]$ . Taking $y_n\in[0,1/2]$ , then

and

$\begin{displaymath} y_{n+1}=2y_n. \end{displaymath}$

(9)

For $y\in [1/2,1]$ ,

and

$\begin{displaymath} y_{n+1}=2-2y_n. \end{displaymath}$

(10)

Combining

$\begin{displaymath} y_n=\cases{ 2y_n & for $y_n \in [0,{\textstyle{1\over 2}}]$\cr 2-2y_n & for $y_n \in [{\textstyle{1\over 2}},1]$,\cr} \end{displaymath}$

(11)

which can be written

$\begin{displaymath} y_n=1-2\vert x_n-h\vert, \end{displaymath}$

(12)

the Tent Map with $\mu=1$ , so the Natural Invariant in

$\begin{displaymath} \rho(y)=1. \end{displaymath}$

(13)

Transforming back to

gives

$\displaystyle \rho(x)$	$\textstyle =$	$\displaystyle \left\vert{dy\over dx}\right\vert \rho(y(x))={2\over \pi} {1\over\sqrt{1-x}} {\textstyle{1\over 2}}x^{-1/2}$
	$\textstyle =$	$\displaystyle {1\over\pi\sqrt{x(1-x)}}.$	(14)

This can also be derived from

$\begin{displaymath} \rho(x)\equiv \lim_{N\to\infty}{1\over N}\sum_{i=1}^N \delta(x_i-x)={1\over \pi\sqrt{x(1-x)}}, \end{displaymath}$

(15)

where $\delta(x)$ is the Delta Function.

See also Logistic Equation