Extended centre manifold of the 2023 Cambridge quadratic-product system

Forset $X=x-a^2$ and $Y=y-a^2$, and append $\dot{a}=0$. Near $(X,Y,a)=(0,0,0)$, the extended centre manifold and its reduced dynamics areThe reduced equation is a subcritical pitchfork with reversed parameter $\mu =-a$.