Extended centre manifold of the 2023 Cambridge quadratic-product system
= Extended centre manifold of the 2023 Cambridge quadratic-product system
For
$$
\dot x=(a^2-x)(a-y^2),
\qquad \dot y=x-y,
$$
set $X=x-a^2$ 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 are
$$
Y=X+aX+O(3),
\qquad
\dot X=-aX+X^3+O(4).
$$
The reduced equation is a subcritical pitchfork with reversed parameter $\mu=-a$.