= Bifurcations of the 2020 Cambridge quintic system
For
$$
\dot x=-x(x^2-2\mu)(x^2-\mu+a),
$$
the 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 bifurcation>[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 pitchfork bifurcation>[imperfect pitchforks] and saddle nodes, so none of these unperturbed bifurcations is structurally stable without the reflection symmetry.
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