= Solution
Representative <normal forms> are $\dot x=\mu+x^2$ for a <saddle-node bifurcation>, $\dot x=\mu x-x^2$ for a <transcritical bifurcation>, and $\dot x=\mu x-x^3$ for a supercritical <pitchfork bifurcation>. Replacing the last cubic term by $+x^3$ gives the subcritical <pitchfork bifurcation>.
Treat $\mu$ as a variable with $\dot\mu=0$. The extended <center manifold> is tangent to $y=0$, so write $y=h(x,\mu)=Ax^2+Bx\mu+C\mu^2+O((|x|+|\mu|)^3)$. Its invariance equation is $h_x\dot x=\dot y$. At quadratic order this reads
$$
(2Ax+B\mu)\mu=(2-A)x^2-Bx\mu-C\mu^2.
$$
Matching coefficients gives $A=2$, $B=-4$, $C=4$. Substitution into the first equation therefore gives
$$
\boxed{y=2x^2-4x\mu+4\mu^2+O(3),\qquad\dot x=\mu+x^2-4x\mu+4\mu^2+O(3).}
$$
Here $O(3)$ denotes total degree at least three in $(x,\mu)$. With $T=t$, $X=x-2\mu$ and $\widetilde\mu=\mu$, this becomes $dX/dT=\widetilde\mu+X^2+O(3)$. Thus $\alpha=1$, $\beta=2$, $\gamma(\mu)=\mu$, and \b[the bifurcation is a saddle-node]. For negative parameter, the negative-$X$ branch is stable and the positive-$X$ branch unstable along the <center manifold>; the transverse <eigenvalue> is near $-1$.
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