Bifurcations of the 2022 Cambridge quadratic-cubic system
= Bifurcations of the 2022 Cambridge quadratic-cubic system
For
$$
\dot x=x(y-k-3x+x^2),
\qquad
\dot y=y(y-1-x),
$$
the invariant center manifold near $(x,y,k)=(0,0,0)$ is $y=0$, with reduced equation $\dot x=x(-k-3x+x^2)$ and hence a <transcritical bifurcation>. Near $(x,y,k)=(1,2,0)$, put $X=x-1$ and $v=y-2-X$. The extended center manifold begins
$$
v=-k+X^2-5Xk+9k^2+O(3),
$$
and its reduced equation is $\dot X=2(X^2-k)+O(3)$, a <saddle-node bifurcation>.