Chaotic behavior of the logistic map at $r=4$. With Sarkovskii's theorem I want to conclude the chaotic behavior of the logistic map $f(x)=r \cdot x(1-x)$. I can't find a value of $x$ which leads to a periodic three orbit. Does anyone know a value of $x$ that leads to a periodic three orbit? If I found such starting point of the iteration, I could use Sarkovskii's theorem to argue the chaotic behavior of the logistic map at $r=4$.
The following pictures (generated with Mathematica) show the graphic iterations of the logistic map for different starting points:
Iteration of the logistic map with starting point $x=0.50000$:

Iteration of the logistic map with starting point $x=0.50001$:

 A: It is well-known that a parametrization of $x_n=\sin^2(\pi \theta_n)$ transforms the dynamic to $\theta_{n+1}=2\theta_n\bmod 1$, the destructive left-shift of the binary digit sequence of $\theta_n$, removing the leading digit in every step. For a period-3 cycle you now need a rational number with a period-3 binary digit sequence like $\theta_0=1/7$, thus take $$x_0=\sin^2(\pi/7).$$
This logic also predicts that using double precision floating point numbers the cycle will deteriorate in about 50 iterations, as then the encoding bits of the initial value are used up, and the remaining contents in $x_n$ will be random bits produced by the floating point truncation errors during the computation.
A: To build up on the Lutz Lehmann's answer, the map $\theta\mapsto \sin^2(\pi\theta)$ exhibits the logistic map $L:x\mapsto 4x(1-x)$ as a $2:1$ factor of the doubling map. Since this latter map has exactly $8-2=6$ points of period $3$, the logistic map $L$ has exactly $3$ points of period $3$, that is, exactly one $3$-cycle. Here is a cobweb diagram exhibiting the cycle:

(The interactive graph is available at: https://www.desmos.com/calculator/ucr4tj7zdq)
