 Solution

For a supercritical pitchfork bifurcation, use $\boxed{x_{n+1}=x_n+\mu x_n-x_n^3}$. The symmetric stable branches $x=\pm \sqrt{\mu }$ appear for $\mu >0$ as the origin becomes unstable. Replacing the cubic minus sign by plus gives a subcritical pitchfork bifurcation.