= Solution
At conduction, $c$ and $e$ have stable <eigenvalues> $-\varpi$ and $-(4-\varpi)\zeta$. The remaining <linearization> on $(a,b,d)$ is
$$
L=\begin{pmatrix}-\sigma&\sigma r&-\sigma\zeta q\\1&-1&0\\1&0&-\zeta\end{pmatrix}.
$$
Its <characteristic polynomial> is $m^3+Am^2+Bm+C$, where
$$
A=\sigma+1+\zeta,\qquad B=\sigma(1-r)+\sigma\zeta(1+q)+\zeta,\qquad C=\sigma\zeta(1+q-r).
$$
Consequently \b[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 gives
$$
\boxed{r_H={ (\sigma+\zeta)(1+\zeta)\over\sigma}+{\zeta(\sigma+\zeta)\over\sigma+1}q,\qquad
\omega^2={\sigma\zeta(1-\zeta)\over\sigma+1}q-\zeta^2.}
$$
For physical $q\ge0$ the oscillatory threshold therefore exists exactly when $0<\zeta<1$ and $q>q_*$. The <conduction double-zero criterion for five-mode magnetoconvection> gives
$$
\boxed{q_*={\zeta(\sigma+1)\over\sigma(1-\zeta)},\qquad r_*=1+q_*,\qquad0<\zeta<1.}
$$
At this <Bogdanov–Takens bifurcation>, the polynomial is $m^2(m+A)$ and the zero <eigenspace> is one-dimensional. For $q<q_*$ conduction first loses stability at the steady threshold; for $q>q_*$ it first loses stability at $r_H<r_s$. For $\zeta\ge1$ 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 $\partial_r(AB-C)=-\sigma(\sigma+1)\ne0$; its nonlinear criticality is not required by this linear onset calculation.
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