 Logistic map two-cycle

For the logistic map $F(u)=\lambda u(1-u)$, factor $F(F(u))-u$ and remove its two fixed points. The remaining quadratic equation is ${\lambda }^2{u}^2-\lambda (\lambda +1)u+\lambda +1=0$. Its distinct roots form a periodic orbit of period two when $\lambda >3$. The periodic-orbit multiplier is $F^{\prime }(u_{-})F^{\prime }(u_{+})=4+2\lambda -{\lambda }^2$, so this orbit is asymptotically stable for $3<\lambda <1+\sqrt{6}$. It is born at a period-doubling bifurcation at $\lambda =3$.