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At conduction, $c$ and $e$ have stable eigenvalues $-\varpi$ and $-(4-\varpi )\zeta$. The remaining linearization on $(a,b,d)$ isIts characteristic polynomial is $m^3+A{m}^2+Bm+C$, whereConsequently the steady threshold is $\boxed{r_s=1+q}$. The symmetry $(a,b,d)\mapsto (-a,-b,-d)$, with $(c,e)$ unchanged, gives a generic pitchfork bifurcation, whose forward or backward direction is determined below. At threshold the other two critical-block eigenvalues satisfy $m^2+Am+B_s=0$, where $B_s=\zeta (\sigma +1)-\sigma (1-\zeta )q$. They are stable if $B_s>0$; if $B_s<0$ there is already one unstable transverse direction. Degenerate zero cubic slope requires higher-order analysis and is not a generic pitchfork.

At a Hopf bifurcation, $AB=C$, $B={\omega }^2>0$. Solving the coefficient equation givesFor physical $q\ge 0$ the oscillatory threshold therefore exists exactly when $0<\zeta <1$ and $q>q_{\ast }$. The conduction double-zero criterion for five-mode magnetoconvection givesAt this Bogdanov–Takens bifurcation, the polynomial is $m^2(m+A)$ and the zero eigenspace is one-dimensional. For $q<q_{\ast }$ conduction first loses stability at the steady threshold; for $q>q_{\ast }$ it first loses stability at $r_H<r_s$. For $\zeta \ge 1$ no positive-$q$ double-zero or oscillatory threshold exists. These conclusions also follow from the Routh-Hurwitz stability criterion, $B,C>0$ and $AB>C$. The Hopf bifurcation is transversal since ${\partial }_r(AB-C)=-\sigma (\sigma +1)\ne 0$; its nonlinear criticality is not required by this linear onset calculation.