= Solution
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 at
$$
\boxed{\mu_G(\lambda)>\mu_H(\lambda)>-1/4,}
$$
tangent 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.
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