 Polar form of the cubic confinement model

A reflection-symmetric planar cubic model can be expressed in polar variables $S=u^2+v^2$, $u=\sqrt{S}\cos \theta$, $v=\sqrt{S}\sin \theta$ as $\dot{S}=2S(\mu +\sin 2\theta -S/2)$ and $\dot{\theta }=\sigma +\cos 2\theta -S/2$. A nonzero equilibrium therefore has $\sin 2\theta =S/2-\mu$, $\cos 2\theta =S/2-\sigma$. Adding their squares gives $S^2/2-(\mu +\sigma )S+{\mu }^2+{\sigma }^2-1=0$. Each positive root $S=\mu +\sigma \pm \sqrt{2-(\mu -\sigma {)}^2}$ determines an antipodal equilibrium pair. The determinant of the polar linearization there is $2S(S-\mu -\sigma )$ and its trace is $2(\mu -S)$. These formulas distinguish folds of nonzero equilibria from symmetry-forced bifurcations at $S=0$.