= Extended centre-manifold reductions of the 2019 Cambridge reflection-symmetric system
{c}
For
$$
\dot x=x+y^2-a,
\qquad
\dot y=y(4x-x^2-a),
$$
the <extended centre manifold for a parameter> gives the local reduced equations
$$
\begin{array}{c|c|c}
(a,x,y)&\text{local parameter and centre coordinate}&\text{reduced equation}\\ \hline
(0,0,0)&\mu=a,\ y&\dot y=3\mu y-4y^3+O(4),\\
(3,3,0)&\mu=a-3,\ y&\dot y=-3\mu y+2y^3+O(4),\\
(4,2,y_0)&\mu=a-4,\ v=y-y_0&\dot v=-y_0(\mu+8v^2)+O(v^3,\mu v),
\end{array}
$$
where $y_0=\pm\sqrt2$. Thus the first two points undergo respectively supercritical and subcritical <symmetry-forced pitchfork bifurcations>, while each of the last two points undergoes a <saddle-node bifurcation>.
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