= Polar form of the cubic confinement model
{title2=$\dot S=2S(\mu+\sin2\theta-S/2)$}
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 $\dot S=2S(\mu+\sin2\theta-S/2)$ and $\dot\theta=\sigma+\cos2\theta-S/2$. A nonzero equilibrium therefore has $\sin2\theta=S/2-\mu$, $\cos2\theta=S/2-\sigma$. 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 $S=0$.
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