Algebraic orbits on period-3 window for the logistic map

Abstract

Algebraic stable and unstable orbits are presented for the famous period-3 window of the logistic map $x_{n+1}=rx_n(1-x_n)$. It is exhibited the general polynomial that gives rise to both stable and unstable period-3 orbits. These orbits are shown for three different fixed control parameter values of $r$: at tangent bifurcation (birth), at super-stability and at ending pitchfork bifurcation (death) of the period-3 window. All orbits are exposed in two different ways: a sum of complex numbers $x_i=a+bc+\overline{bc}$, as proposed by Gordon (Math Mag 69:118–120, 1996), and via Euler’s formula $x_i=a+2|b|\cos (\theta )$. The algebraic expressions of $a, b, c, |b|$ and $\theta$ are given for each $r$ value for both stable and unstable orbits, as well as their numerical values and the Lyapunov exponent. It is shown that $a$ and $|b|$ are statistical quantities of the orbits.