= Solution
Use local coordinates $q=\theta-\pi/2$, $p=\phi$ at $P_1$. The <Jacobian matrix>, its <trace> and its <determinant> are
$$
J=\begin{pmatrix}\lambda-1&-2\\1-\kappa&-2\end{pmatrix},\qquad T=\lambda-3,\qquad D=2(2-\lambda-\kappa).
$$
Consequently $P_1$ is attracting when $T<0,D>0$, repelling when $T>0,D>0$, and a <saddle equilibrium> when $D<0$. A transition between a <node> and a <focus> at $T^2=4D$ changes the linear geometry but is not a <bifurcation> of the local topological <phase portrait>.
The <stationary bifurcation> curve is $\lambda+\kappa=2$. To identify it, write $C=\cos2\phi$, $S=\sin2\phi$. At a non-equatorial <equilibrium point>, eliminating $S$ gives
$$
C=\frac{\lambda+\kappa}{2},\qquad \cos\theta=\frac{2S}{\kappa-\lambda}.
$$
Near $(3,-1)$, if $d=2-\lambda-\kappa>0$, these yield $p=\pm\sqrt d/2+O(d^{3/2})$ and $q=p+o(\sqrt d)$. The two branches are related by the local reflection $(q,p)\mapsto(-q,-p)$. Their <determinants> are $-4d+o(d)$, so they are <saddle equilibria>. This is a <pitchfork bifurcation>; on the portion with $T<0$ the saddle branches lie on the side where $P_1$ is stable, giving a <subcritical pitchfork bifurcation>.
A <Hopf bifurcation> requires $T=0,D>0$, hence
$$
\boxed{\lambda=3,\quad\kappa<-1.}
$$
The oscillation frequency is $\omega=\sqrt{-2(1+\kappa)}$. At the intersection $\boxed{(\lambda,\kappa)=(3,-1)}$, $J\ne0$ but $J^2=0$: there is a double zero <eigenvalue> with one <eigenvector>. The map from $(\lambda,\kappa)$ to $(T,D)$ has nonzero <Jacobian determinant>, so two independent parameter conditions produce this <codimension-two bifurcation>. Reflection removes quadratic terms; the appropriate initial comparison is with a <reflection-symmetric cubic double-zero normal form>, subject to the cubic degeneracy checked below.
For completeness, the printed <vector field> has a subcritical, rather than supercritical, <Hopf bifurcation>. In the mechanical coordinates $x=p$, $y=\dot p$, at $\lambda=3$ its cubic acceleration has $x^2y$ coefficient $b=6(\kappa-3)(\kappa+1)/(\kappa-1)^2$ and $y^3$ coefficient $d_3=-2/(\kappa-1)^2$. For a small harmonic oscillation, the averaged cubic change of $H=(y^2+\omega^2x^2)/2$ is $\omega^2A^4(b+3d_3\omega^2)/8$, where
$$
b+3d_3\omega^2=\frac{6(\kappa+1)}{\kappa-1}>0.
$$
Thus a small unstable <periodic orbit> lies on the attracting side of the <Hopf bifurcation>. This calculation matters when comparing the original equations with the later assumed <normal form>.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2003/iii/paper-59-bifurcation-diagrams.png]
{title=Local bifurcation curves for the spherical system, rotating convection, and the fold–Hopf system}
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