= Solution
Let the <equilibrium> be $A_*=Re^{i\theta}$ and write $\delta A=e^{i\theta}(u+iv)$. The <linearization> of the cubic term gives $\delta(|A|^2A)=2s\delta A+A_*^2\delta\overline A$. Therefore the real <stability matrix> is
$$
J=\begin{pmatrix}\mu-3s&-\Lambda\\\Lambda&\mu-s\end{pmatrix}.
$$
Its <eigenvalues> obey
$$
\boxed{\sigma^2+2(2s-\mu)\sigma+(3s-\mu)(s-\mu)+\Lambda^2=0,\qquad \sigma_\pm=\mu-2s\pm\sqrt{s^2-\Lambda^2}.}
$$
The square root may be imaginary; stability uses real parts. The <determinant> $D$ is exactly $F_s$, so a zero <eigenvalue> occurs only at the turning points derived above. Away from the cusp those are ordinary <saddle-node bifurcations>; the parameterization joining point has $D>0$.
At a candidate <Hopf bifurcation>, the <trace> vanishes: $\mu_H=2s_H$. Substitution into the steady cubic gives $s_H(s_H^2+\Lambda^2)=1$, which has one positive root. The pair is genuinely imaginary only if
$$
\omega_H^2=D_H=\Lambda^2-s_H^2=\frac1{s_H}-2s_H^2>0.
$$
Thus $s_H<2^{-1/3}$ and, equivalently,
$$
\boxed{\Lambda>2^{-1/3},\qquad \mu_H=2s_H,\qquad \omega_H=\sqrt{\Lambda^2-s_H^2}.}
$$
This direct calculation proves the threshold without needing the additional monotonicity assumption. The equality gives a double-zero <eigenvalue>, not an ordinary <Hopf bifurcation>. Along a fixed-$\Lambda$ branch, implicit differentiation gives $ds/d\mu=-2s(\mu-s)/D$; at this crossing $d\operatorname{Re}\sigma/d\mu=1+4s_H^2/D_H>0$. A nonzero nonlinear Hopf coefficient then gives the usual <Hopf bifurcation>.
The <fold and Hopf thresholds of a directly forced cubic amplitude> divide the diagrams into three regimes. At noncritical points the two <eigenvalues>, counted with multiplicity by the signs of their real parts, are classified as follows:
* For $\Lambda>\sqrt3/2$, there is one <equilibrium> for every $\mu$. It has two stable <eigenvalues> for $\mu<\mu_H$ and two unstable <eigenvalues> for $\mu>\mu_H$.
* For $\sqrt3/2>\Lambda>2^{-1/3}$, the largest-amplitude branch has two stable <eigenvalues>; the intermediate-amplitude branch has one stable and one unstable <eigenvalue>. The smallest-amplitude branch emerging from the lower fold has two stable <eigenvalues> up to its Hopf crossing, and two unstable <eigenvalues> beyond it. The relative location of the Hopf point and the other fold in $\mu$ is not fixed throughout this interval.
* For $0<\Lambda<2^{-1/3}$, the largest-amplitude branch again has two stable <eigenvalues>, the intermediate branch has one of each, and the smallest-amplitude branch has two unstable <eigenvalues>. The trace-zero point lies on the saddle segment and is not a <Hopf bifurcation> because its <determinant> is negative.
At a fold one <eigenvalue> is zero; at a Hopf point the pair has zero real part. These nonhyperbolic points are marked separately in the sketches.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2003/iii/paper-60-forced-amplitude-branches.png]
{title=Steady amplitudes of the moving forced pattern, with folds, Hopf points and stable/unstable <eigenvalue> counts in all three regimes}
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