= Three-amplitude vertical-field magnetoconvection
For one stress-free planar roll mode, thermal-time rescaling reduces the linear dynamics to $a'=\sigma(-a+rb-\zeta qc)$, $b'=a-b$, $c'=a-\zeta c$. Its characteristic polynomial is $(s+\sigma)(s+1)(s+\zeta)-\sigma r(s+\zeta)+\sigma\zeta q(s+1)$. For $\sigma,\zeta>0$ and $q\ge0$, steady marginality is $r_s=1+q$. Oscillatory marginality is $r_H=(\sigma+\zeta)(1+\zeta)/\sigma+\zeta(\sigma+\zeta)q/(\sigma+1)$, with $\omega_H^2=\zeta[\sigma(1-\zeta)q/(\sigma+1)-\zeta]$. A genuine <Hopf bifurcation> onset requires $\omega_H^2>0$; a formal imaginary-root formula with negative frequency squared is not an oscillation. The <Routh-Hurwitz stability criterion> selects the first crossing, while nonlinear coefficients determine criticality.
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