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.