= Stationary branch of five-mode magnetoconvection
{title2=$r=1+a^2+\frac{q\mu(1+a^2)[\mu+(4-\varphi)a^2]}{(\mu+a^2)^2}$}
For positive diffusivity ratio $\zeta$ and $0<\varphi<4$, define $\mu=(4-\varphi)\zeta^2/\varphi$. 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 $\varphi q(2-\varphi)+(4-\varphi)\zeta^2(1+q)$. 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.
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