= Solution
Let $q=r_\Omega^2$ and $h=\varpi$. The $c,e$ <eigenvalues> at the conductive <equilibrium point> are $-h$ and $-(4-h)\sigma$, both negative since $0<h<4$ and $\sigma>0$. The coupled $(a,b,d)$ <Jacobian matrix> has characteristic polynomial
$$
P(s)=(s+\sigma)^2(s+1)-\sigma r(s+\sigma)+\sigma^2q(s+1)
=s^3+A_1s^2+A_2s+A_3,
$$
where
$$
A_1=1+2\sigma,\qquad A_2=\sigma^2+2\sigma-\sigma r+\sigma^2q,\qquad A_3=\sigma^2(1+q-r).
$$
A simple zero <eigenvalue> occurs on $\boxed{r_P=1+q}$. The reflection $(a,b,d)\mapsto(-a,-b,-d)$, with $c,e$ fixed, makes its generic <steady-state bifurcation> a <pitchfork bifurcation>.
For a nonzero imaginary pair, substitute $s=i\omega$ into $P$: the real and imaginary parts require $A_3=A_1A_2$ and $\omega^2=A_2>0$. Hence
$$
\boxed{r_H=2(1+\sigma)+\frac{2\sigma^2}{1+\sigma}q,\qquad
\omega_H^2=\sigma^2\left(\frac{1-\sigma}{1+\sigma}q-1\right).}
$$
With $0<\sigma<1$, this <Hopf bifurcation> curve is present only for $q>q_{TB}=(1+\sigma)/(1-\sigma)$. At its intersection with the <pitchfork bifurcation> curve,
$$
\boxed{q_{TB}=\frac{1+\sigma}{1-\sigma},\qquad r_{TB}=\frac2{1-\sigma},}
$$
and $P(s)=s^2(s+1+2\sigma)$. The zero <eigenvalue> has geometric multiplicity one: its <eigenvector> has $b=a$, $d=a/\sigma$. Thus this is a reflection-symmetric <Bogdanov–Takens bifurcation>, rather than two independent zero modes.
The <Routh-Hurwitz stability criterion> gives the stable trivial state below $r_P$ for $q<q_{TB}$ and below $r_H$ for $q>q_{TB}$. The rest of the line $r_P$, beyond $q_{TB}$, is a stationary <bifurcation> of an already unstable trivial state. Likewise the extension of $r_H$ below $q_{TB}$ is only an algebraic line, since its putative frequency is not real. The middle panel of the <bifurcation diagram> in part 1(b) distinguishes the actual curves from this extension.
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