Bifurcations of the 2020 Cambridge quintic system

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.