Degenerate pitchfork in the cubic confinement model ($|u|+|v|\asymp |\delta \mu {|}^{1/4}$)

The polar form of the cubic confinement model has an origin pitchfork on ${\mu }^2+{\sigma }^2=1$ when $\mu \ne 0$. Its small squared amplitude satisfies $S\sim 2\mu \delta \mu /(\mu +\sigma )$ along fixed $\sigma$. Thus the leading cubic coefficient vanishes when $\mu +\sigma =0$, at $(\sigma ,\mu )=(\pm 1/\sqrt{2},\mp 1/\sqrt{2})$. 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.