Three-amplitude vertical-field magnetoconvection

For one stress-free planar roll mode, thermal-time rescaling reduces the linear dynamics to $a^{\prime }=\sigma (-a+rb-\zeta qc)$, $b^{\prime }=a-b$, $c^{\prime }=a-\zeta c$. Its characteristic polynomial is $(s+\sigma )(s+1)(s+\zeta )-\sigma r(s+\zeta )+\sigma \zeta q(s+1)$. For $\sigma ,\zeta >0$ and $q\ge 0$, 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.