Secondary steady and Hopf thresholds of a resonant pure mode ($r_S=-\frac{|d{|}^2}{2\mathrm{Re}(\beta \overline{d})},r_H=-\frac{d_R}{{\beta }_R}$)

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 $-2(d_R+{\beta }_{R}r)$ and determinant $|d{|}^2+2rQ$, where $Q=\mathrm{Re}(\beta \overline{d})$. A positive steady threshold requires $Q<0$; a simple zero eigenvalue also requires $Q\ne Q_{\ast }={\beta }_R|d{|}^2/(2d_R)$. A Hopf bifurcation requires $Q>Q_{\ast }$, giving a positive determinant at $r_H$. For $Q<Q_{\ast }$ the first instability is steady, producing reflection-related mixed phase-locked periodic branches. For $Q>Q_{\ast }$ it is Hopf, producing modulated oscillations, generically quasiperiodic in the original variables. Equality is a double-zero degeneracy.