Solution
= Solution
Put $x=1+u$, $y=v$, and $μ=-1+ν$, adjoining $\dot ν=0$. Then
$$
\dot u=v(ν+u+v),\qquad \dot v=-u-v-u^2.
$$
The zero <eigenvalue> has centre direction $v=-u$. Solving the <center manifold> invariance equation gives
$$
v=-u-u^2-uν+O(|(u,ν)|^3).
$$
Substitution yields the reduced equation
$$
\dot u=u(u^2-ν)+O(|(u,ν)|^4).
$$
Its equilibria are $u=0$ and $u=±\sqrt ν+O(ν)$ for $ν>0$, exactly matching the three branches meeting at the pitchfork in part (b).