= Degenerate pitchfork in the cubic confinement model
{title2=$|u|+|v|\asymp|\delta\mu|^{1/4}$}
= Degenerate pitchforks in the cubic confinement model
{synonym}
The <polar form of the cubic confinement model> has an origin pitchfork on $\mu^2+\sigma^2=1$ when $\mu\ne0$. Its small squared amplitude satisfies $S\sim2\mu\delta\mu/(\mu+\sigma)$ along fixed $\sigma$. Thus the leading cubic coefficient vanishes when $\mu+\sigma=0$, at $(\sigma,\mu)=(\pm1/\sqrt2,\mp1/\sqrt2)$. At either point the exact radius equation gives $\delta\mu=-S^2/(4\mu_0)+o(S^2)$, so amplitude scales as $|\delta\mu|^{1/4}$. The lower point has a stabilizing quintic coefficient $1/(4\mu_0)<0$; folds of nonzero equilibrium pairs meet the pitchfork curve there. The upper point has the opposite quintic sign and an already unstable transverse eigenvalue.
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