= Conduction double-zero criterion for five-mode magnetoconvection
{title2=$q_*={\zeta(\sigma+1)\over\sigma(1-\zeta)},\quad r_*=1+q_*$}
The conduction linearization has two negative <eigenvalues> and a cubic factor $s^3+As^2+Bs+C$, where $A=\sigma+1+\zeta$, $B=\sigma(1-r)+\sigma\zeta(1+q)+\zeta$, $C=\sigma\zeta(1+q-r)$. A double zero requires $B=C=0$, giving the displayed <Bogdanov–Takens bifurcation> location for $0<\zeta<1$. The remaining cubic eigenvalue is $-A$. The imaginary-pair condition $AB=C$ yields $r_H=(\sigma+\zeta)(1+\zeta)/\sigma+\zeta(\sigma+\zeta)q/(\sigma+1)$ and $\omega^2=\sigma\zeta(1-\zeta)q/(\sigma+1)-\zeta^2$. Thus the oscillatory onset exists precisely for $q>q_*$ when $0<\zeta<1$.
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