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Forthe equilibrium branches are $x=0$, $x=\pm \sqrt{2\mu }$, and $x=\pm \sqrt{\mu -a}$ wherever the square roots are real. If $a<0$, pitchforks occur at $\mu =a$ and $\mu =0$, and the two nonzero branch pairs meet in two transcritical bifurcations at $\mu =-a$. If $a=0$, all five branches meet in one degenerate fifth-order bifurcation. If $a>0$, separate pitchforks occur at $\mu =0$ and $\mu =a$. A constant perturbation produces imperfect pitchforks and saddle nodes, so none of these unperturbed bifurcations is structurally stable without the reflection symmetry.

After adding $ϵ>0$, the equilibrium equation isThe 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 bifurcations. The lower row shows this unfolding for a representative small positive $ϵ$; the upper row gives the unperturbed diagrams.

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,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.