= Logistic map two-cycle
{title2=$u_\pm=(\lambda+1\pm\sqrt{(\lambda-3)(\lambda+1)})/(2\lambda)$}
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 $\lambda^2u^2-\lambda(\lambda+1)u+\lambda+1=0$. Its distinct roots form a <periodic orbit> of period two when $\lambda>3$. The <periodic-orbit multiplier> is $F'(u_-)F'(u_+)=4+2\lambda-\lambda^2$, so this orbit is <asymptotically stable> for $3<\lambda<1+\sqrt6$. It is born at a <period-doubling bifurcation> at $\lambda=3$.
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