Algebraic period 3 orbits: rotation on complex plane and statistical quantities.

For the period-$3$ window of the logistic map $x_{n+1}=rx_n(1-x_n)$ it is presented algebraic orbits, for both stable and unstable ones.
From the solution of $x_{n+3}=x_n$ it is obtained the polynomial that rules the periodic orbits inside this window.
The roots of this polynomial are the orbits, and they are functions of the fixed parameter value $r$.
As $r$ is increased, the value of the roots are modified: some increase and the others decrease.
For the same branch of the three possible orbits, the value of the roots present opposite behavior with respect to stable and unstable orbits.
The roots of the polynomial, i.e., the orbits, are presented in two different ways: a sum of complex numbers $x_i=a+bc+\overline{bc}$, and via Euler's formula $x_i=a+2|b|\cos(\theta)$ -- the overbar indicates complex conjugation.
The algebraic orbits are obtained for three different fixed control parameter values of $r$: at tangent bifurcation (birth), at super-stability and at ending pitchfork bifurcation (death).
The algebraic expressions of the constants $a$, $b$, $c$, $|b|$ and $\theta$ are given for each $r$ value for both stable and unstable orbits.
It is shown that $a$ and $|b|$ are statistical quantities of the orbits, again, both stable and unstable ones.
Finally, the numerical values of the orbits and the constants obtained are summarized in a table and, numerically, it is shown the behavior of each orbit, constant and Lyapunov exponent for each orbit.