Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 4 / 31A / a / i / Solution

For $b=0,a>0$, the nonzero branches $x=\pm \sqrt{(1-\mu )/a}$ exist for $\mu <1$ and have multiplier $3-2\mu >1$, so both repel. Zero changes from attracting below one to repelling above one: the bifurcation at one is a subcritical pitchfork bifurcation. At minus one its multiplier crosses $-1$, producing a supercritical period-doubling bifurcation; the newly stable small two-cycle lies on the side $\mu <-1$.