Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 31A / c / Solution

Put $x=1+u$, $y=v$, and $\mu =-1+\nu$, adjoining $\dot{\nu }=0$. ThenThe zero eigenvalue has centre direction $v=-u$. Solving the center manifold invariance equation givesSubstitution yields the reduced equationIts equilibria are $u=0$ and $u=\pm \sqrt{\nu }+O(\nu )$ for $\nu >0$, exactly matching the three branches meeting at the pitchfork in part (b).