= Cubic centre-manifold reduction of porous magnetoconvection
{title2=$\dot a=\frac{\sigma\zeta}{\zeta-\sigma q}\mu a+\frac{3\sigma q}{\zeta(\zeta-\sigma q)}a^3+\cdots$}
For <three-mode porous magnetoconvection>, set $\mu=r-1-q$ and assume $0<q<\zeta/\sigma$. The transverse <eigenvalues> are negative. Centre-manifold invariance gives
$$
c=\frac{3a^2}{\zeta^2}+\cdots,\qquad
b=\frac a\zeta-\frac{\sigma\mu a}{\zeta(\zeta-\sigma q)}-\frac{3a^3}{\zeta^2(\zeta-\sigma q)}+\cdots.
$$
Substitution into the first equation gives the displayed positive linear and cubic coefficients. Thus the bifurcation is a <subcritical pitchfork bifurcation>: the origin is stable for $\mu<0$ and unstable for $\mu>0$, while the small nonzero branches at $a^2=-\zeta^2\mu/(3q)$ are unstable. Simply setting $\dot b=0$ misses the time-derivative contribution responsible for the common factor $(1-\sigma q/\zeta)^{-1}$.
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