Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 56 / 2 / b

At $\mu =-1/4$, the two branches meet at $(1/2,1/4)$. The eigenvalues there are $0$ and $1/2-\lambda$. For $\lambda \ne 1/2$, the nonzero quadratic term and transverse stability identify a saddle-node bifurcation. If $\lambda >1/2$, it creates a sink equilibrium and a saddle equilibrium as $\mu$ increases; if $\lambda <1/2$, it creates a source equilibrium and a saddle equilibrium. At $\lambda =1/2$, the zero eigenvalue is double and the ordinary saddle-node description needs the two-parameter unfolding.

At $\mu =0$, the origin and lower branch cross. Since $s_{-}=-\mu +O({\mu }^2)$, the branches cross transversely in the $(\mu ,u)$ diagram. The origin changes from a sink equilibrium to a saddle equilibrium as $\mu$ increases through zero; the lower branch changes from a positive-$u$ saddle equilibrium to a negative-$u$ sink equilibrium. This is a transcritical bifurcation. A local centre-manifold calculation also gives $\dot{u}=\mu u+u^2+\cdots$, because $v=O(u^2)$ and the transverse eigenvalue is $-\lambda <0$.

The figure uses a representative $\lambda >1/2$; the upper branch's additional Hopf change is indicated. For $\lambda <1/2$ the upper branch is unstable everywhere on its existence interval. The exact trace criterion in part (a) specifies stability for every fixed positive $\lambda$.