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Logistic Equation r=4

With $r=4$, 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)

so
\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 $s=0$ and
\begin{displaymath}
y_{n+1}=2y_n.
\end{displaymath} (9)

For $y\in [1/2,1]$, $s=1$ 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 $y$ is
\begin{displaymath}
\rho(y)=1.
\end{displaymath} (13)

Transforming back to $x$ 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



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© 1996-9 Eric W. Weisstein
1999-05-25