Stationary branch of five-mode magnetoconvection ($r=1+a^2+\frac{q\mu (1+a^2)[\mu +(4-\phi )a^2]}{(\mu +a^2{)}^2}$)

For positive diffusivity ratio $\zeta$ and $0<\phi <4$, define $\mu =(4-\phi ){\zeta }^2/\phi$. The nonzero steady branch has $b=a/(1+a^2)$, $c=a^2/(1+a^2)$, $d=\mu a/[\zeta (\mu +a^2)]$, and $e=a^2/(\mu +a^2)$. Substitution yields the displayed branch through $r=1+q$. Its initial slope in $a^2$ has the sign of $\phi q(2-\phi )+(4-\phi ){\zeta }^2(1+q)$. Positive slope is a supercritical pitchfork bifurcation direction, negative slope a subcritical pitchfork bifurcation direction. This geometric criterion alone does not assert stability against every perturbation.

Put $m=(4-\phi ){\zeta }^2/\phi$. The derivative of the stationary branch of five-mode magnetoconvection at $a^2=0$ is $r_2=1+q+q(2-\phi )/m$. For fixed positive $q$ and small $\zeta$, it changes from a forward to a backward pitchfork bifurcation direction at the displayed value. Expanding the exact numerator $q\phi (2-\phi )+(4-\phi ){\zeta }^2(1+q)$ gives this result. The expansion in amplitude requires $a^2\ll m$; branch direction alone does not determine stability against all other modes.