= Solution
The <centre manifold theorem> gives, near an equilibrium of a sufficiently smooth system, a local invariant <manifold> tangent to the <generalized eigenspaces> with zero-real-part <eigenvalues>, expressible as a <graph of a function> over them. Its reduced dynamics govern local equilibria and their stability in the presence of strictly stable transverse directions. The <manifold> need not be unique or analytic; finite Taylor jets can nevertheless be determined from invariance. Treating a parameter as a variable with $\dot\mu=0$ gives the <extended centre manifold for a parameter>.
At $r=1$ the linearized $(x,y)$ block has <eigenvalues> $0,-2$, and the $z$ direction has <eigenvalue> $-1$. Thus the origin is a <nonhyperbolic equilibrium>, with \b[stable <dimension> two and non-extended centre <dimension> one]. The extended centre <dimension> is two. The transformed equations are
$$
\begin{aligned}
\dot u&=\tfrac12\mu(u+v)+\tfrac a2(u+v)^3-\tfrac12(u-v)z,\\
\dot v&=-2v-\tfrac12\mu(u+v)+\tfrac a2(u+v)^3+\tfrac12(u-v)z,\\
\dot z&=u^2-v^2-z,\qquad\dot\mu=0.
\end{aligned}
$$
The <graph of a function> satisfies $V(0,0)=Z(0,0)=0$ and $DV(0,0)=DZ(0,0)=0$, with invariance equations $V_u\dot u=\dot v$ and $Z_u\dot u=\dot z$. Symmetry permits $V$ odd and $Z$ even in $u$. Giving $u$ weight one and $\mu$ weight two, comparison yields
$$
Z=u^2+O_{\rm w}(4),\qquad
V=-\frac14\mu u+\frac{a+1}{4}u^3+O_{\rm w}(5),\qquad
\boxed{\dot u=\frac12\mu u+\frac{a-1}{2}u^3+O_{\rm w}(5).}
$$
Here $O_{\rm w}(j)$ means terms of weighted order at least $j$ under the stated scaling. If $a<1$, the <pitchfork bifurcation> is supercritical: stable branches $u\sim\pm\sqrt{\mu/(1-a)}$ appear for $\mu>0$ as the origin loses stability. If $a>1$, it is a <subcritical pitchfork bifurcation>: unstable branches occur for $\mu<0$, while the origin is stable there.
For $a=1$, assign $\mu$ weight four. The leading cubic terms in $\dot u$ cancel. The invariance equations now give $V=u^3/2+O_{\rm w}(5)$ and $Z=u^2+0\,u^4+O_{\rm w}(6)$: $Z_u\dot u$ first contributes at weight six and $v^2$ also first contributes there. Substitution then gives
$$
\boxed{\dot u=\frac12\mu u+u^5+O_{\rm w}(7),\qquad
u\sim\pm(-\mu/2)^{1/4}\quad(\mu<0).}
$$
The <derivative> along the nonzero branches is $-2\mu>0$, so they are unstable. The origin is stable for $\mu<0$, unstable for $\mu>0$, and nonlinearly unstable at $\mu=0$ because the leading term is $+u^5$. This is a degenerate subcritical <pitchfork bifurcation>.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2017/ii/paper-3-bifurcation.png]
{title=<Bifurcation diagram> for the equilibrium branches}
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