= Solution
Write $h=\mu^2+\varepsilon$. The entire unfolded diagram is obtained from $g(x)=h$, with the same <dynamical stability> sign $-x(x+2)$ as before. A <saddle-node bifurcation> at $x=0$ requires $h=0$, while a <saddle-node bifurcation> at $x=-2$ requires $h=4/3$.
For $0<\varepsilon<4/3$, \b[the crossing separates into a positive stable branch and a negative unstable branch], with no <saddle-node bifurcation> near $\mu=0$. Three <equilibria> exist for $|\mu|<\sqrt{4/3-\varepsilon}$, and the two negative branches annihilate at the outer <saddle-node bifurcations> $\mu=\pm\sqrt{4/3-\varepsilon}$. Outside that interval only the positive stable <equilibrium> remains. If $\varepsilon>4/3$, the diagram has just that one branch everywhere; equality is the limiting <saddle-node bifurcation> contact at $\mu=0,x=-2$.
For $\varepsilon<0$, let $d=-\varepsilon>0$. \b[Two inner <saddle-node bifurcations> occur at $\mu=\pm\sqrt d,x=0$], and two outer <saddle-node bifurcations> at $\mu=\pm\sqrt{4/3+d},x=-2$. For $|\mu|<\sqrt d$ there is a single stable <equilibrium> below $-3$; for $\sqrt d<|\mu|<\sqrt{4/3+d}$ there are three <equilibria> with stable–unstable–stable ordering; outside the outer <saddle-node bifurcations> only the positive stable <equilibrium> remains. The preceding figure shows representative small unfoldings. In particular this unrestricted perturbation destroys the exact <transcritical bifurcation> crossing.
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