Solution
= Solution
For $b=1$, set $A=a+1/4$ and $X=x-1/2$. The centre manifold has $y=A-X^2+\cdots$, so $\dot X=A-X^2+\cdots$. This is a saddle-node at $a=-1/4$, with two branches for $a>-1/4$.
= Solution
For $b=1$, set $A=a+1/4$ and $X=x-1/2$. The centre manifold has $y=A-X^2+\cdots$, so $\dot X=A-X^2+\cdots$. This is a saddle-node at $a=-1/4$, with two branches for $a>-1/4$.