= Oscillatory threshold of a thermosolutal Lorenz model
{title2=$r_H=1+\tau(\sigma+1+\tau)/\sigma+r_s(\sigma+\tau)/(\sigma+1)$}
Linearization of the <Lorenz model of thermosolutal convection> at conduction yields $(\lambda+\sigma)(\lambda+1)(\lambda+\tau)-\sigma r(\lambda+\tau)+\sigma r_s(\lambda+1)=0$. A <Hopf bifurcation> requires the cubic coefficient identity $a_3=a_1a_2$ and positive frequency squared. Its threshold is the stated $r_H$, with $\omega^2=\sigma r_s(1-\tau)/(\sigma+1)-\tau^2>0$. If $r_s=\beta^2\tau$ and $\delta=\sigma/(1+\sigma)$, then $r_H=1+\tau/\delta+\beta^2\tau\delta+O(\tau^2)$. Comparing with the <steady fold of a thermosolutal Lorenz model> gives a leading nonnegative difference $\tau(\beta\delta-1)^2/\delta$ for $\beta\ge0$.
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