 Cubic map period-two branches

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 $2<\mu <\sqrt{5}$ and undergo a period-doubling bifurcation at $\sqrt{5}$. At $\mu =2$ these points coincide with fixed points, not genuine two-cycles.