= Secondary steady and Hopf thresholds of a resonant pure mode
{title2=$r_S=-\frac{|d|^2}{2\operatorname{Re}(\beta\overline d)},\quad r_H=-\frac{d_R}{\beta_R}$}
In the symmetric resonant cubic <amplitude equation>, put $A=\sqrt r\,e^{i\Omega t}$, $B=C e^{i\Omega t}$, with $d_R>0$, $\beta_R<0$. The transverse equation is $\dot C=-(d+\beta r)C-\beta r\overline C$. Its real matrix has trace $-2(d_R+\beta_Rr)$ and determinant $|d|^2+2rQ$, where $Q=\operatorname{Re}(\beta\overline d)$. A positive steady threshold requires $Q<0$; a simple zero <eigenvalue> also requires $Q\ne Q_*=\beta_R|d|^2/(2d_R)$. A <Hopf bifurcation> requires $Q>Q_*$, giving a positive determinant at $r_H$. For $Q<Q_*$ the first instability is steady, producing reflection-related mixed phase-locked periodic branches. For $Q>Q_*$ it is Hopf, producing modulated oscillations, generically quasiperiodic in the original variables. Equality is a double-zero degeneracy.
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