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

The global curve is a homoclinic connection to the lower positive equilibrium, which is the saddle equilibrium created at the saddle-node bifurcation. Near $\lambda =1/2$ on the side $\lambda >1/2$, place it attangent to $\mu =-1/4$ at the double-zero point. The stable periodic orbit occupies the strip ${\mu }_H<\mu <{\mu }_G$ and approaches the saddle equilibrium on the upper boundary. The dashed curve in the two-parameter figure indicates this local placement; its drawn numerical coefficient is schematic, not a computed global-bifurcation formula.

This placement is consistent with the supplied normal form. Its centre equilibrium has a supercritical Hopf bifurcation, and its saddle equilibrium has negative trace on the relevant loop branch, allowing a stable periodic orbit to disappear in the saddle equilibrium loop. No global-curve calculation is required here.