= Cubic map period-two branches
{title2=$F(x)=x(\mu-x^2)$}
The displayed map has a symmetric <periodic orbit> $\{\pm\sqrt{\mu+1}\}$ for $\mu>-1$, with multiplier $(-2\mu-3)^2>1$. For $\mu>2$, put $s_\pm=(\mu\pm\sqrt{\mu^2-4})/2$. The two additional period-two orbits are $\{\sqrt{s_+},\sqrt{s_-}\}$ and their negatives. Their multiplier is $9-2\mu^2$, so they are stable for $2<\mu<\sqrt5$ and undergo a <period-doubling bifurcation> at $\sqrt5$. At $\mu=2$ these points coincide with fixed points, not genuine two-cycles.
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