 Pitchfork reversal near geometric ratio two in magnetoconvection

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.