Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 3 / 7A / i / Solution

A representative normal form for a saddle-node bifurcation of a map is $\boxed{x_{n+1}=x_n+\mu -x_n^2}$. Two fixed points $x=\pm \sqrt{\mu }$ appear for $\mu >0$; near the bifurcation the positive branch is stable and the negative branch unstable.