Past exam of the topics in applied mathematics course of the University of Oxford / 2007 / B / 5 / a / Solution

A saddle-node bifurcation creates or destroys two equilibria when a real eigenvalue passes through zero with a nonzero quadratic normal-form term. With a parameter orientation chosen so the pair appears for $\mu >0$, useThere are no equilibria for $\mu <0$, a double nonhyperbolic equilibrium at zero, and equilibria $q=\pm \sqrt{\mu }$ for $\mu >0$. The positive branch is stable and the negative branch unstable in the active direction; additional hyperbolic directions can make them a node and a saddle equilibrium. “Supercritical” for a fold depends on this chosen parameter orientation; unlike the pitchfork and Hopf cases it is not an intrinsic distinction between the stability of two possible cubic branches.