= Solution
After adding $\epsilon>0$, the equilibrium equation is
$$
x^5+(a-3\mu)x^3+2\mu(\mu-a)x-\epsilon=0.
$$
The constant term breaks the reflection symmetry $x\mapsto-x$. Consequently each symmetry-forced pitchfork becomes an <imperfect pitchfork bifurcation>, while each crossing that produced a <transcritical bifurcation> is disconnected. The remaining generic turning points are <saddle-node bifurcation>[saddle-node bifurcations]. The lower row shows this unfolding for a representative small positive $\epsilon$; the upper row gives the unperturbed diagrams.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2020/ii/paper-1-bifurcation-diagrams.png]
{title=<Bifurcation diagrams> for the equilibrium branches}
For $a<0$, both pitchforks and both transcritical crossings are destroyed. For $a=0$, the simultaneous fifth-order meeting splits into separated folds. For $a>0$, both pitchforks become imperfect. Hence, under the allowed constant perturbation,
$$
\boxed{\text{none of the original bifurcations is structurally stable}.}
$$
This is the <structural stability of a bifurcation> distinction between the nongeneric pitchfork and transcritical diagrams and the generic saddle-node diagrams left by their unfoldings.
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