= Solution
A Hopf crossing of the transverse <stability matrix> requires zero trace and strictly positive determinant. Since $\beta_R<0$, the trace condition gives the positive threshold
$$
\boxed{r_H=R^2=-\frac{d_R}{\beta_R}.}
$$
The determinant there is
$$
D_H=|d|^2-\frac{2d_RQ}{\beta_R}.
$$
Thus the required <coefficient> condition and the rotating-frame Hopf frequency are
$$
\boxed{Q>Q_*:=\frac{\beta_R|d|^2}{2d_R},\qquad \omega_H=\sqrt{D_H}>0.}
$$
The trace derivative $-2\beta_R$ is strictly positive, so with $d,\beta$ fixed the <real parts> cross zero with nonzero speed. Under this condition the determinant stays positive from $r=0$ to $r_H$; the pure branch first loses <linear stability> through this <Hopf bifurcation>. When $Q<Q_*$ its first instability is instead the steady one of part (a), and $D_H<0$ prevents a Hopf pair at the trace-zero point. Equality gives the already noted double-zero degeneracy.
Generically the new branch is periodic in the symmetry-reduced rotating frame, with a small oscillating odd-mode <amplitude> and a modulation of the even-mode <amplitude>. Reconstructing the common temporal <phase> combines the modulation frequency with the carrier frequency. The original <amplitudes> therefore describe \b[modulated periodic oscillations, generically a two-frequency quasiperiodic state]. If the two frequencies are rationally related, the reconstructed motion can be periodic instead. The spatial pattern no longer has a fixed mixture of the even and odd modes: their relative <amplitudes> and <phase> oscillate. As with the steady case, nonlinear nondegeneracy is needed for an ordinary Hopf branch, and its criticality or stability cannot be inferred from the trace and determinant alone.
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