Connecting pitchfork and transcritical branches in a quadratic-product flow ($\dot{x}=x(\mu -x+y^2),\dot{y}=y(1-x-y^2)$)

At $(x,y)=(\mu ,0)$ near $\mu =1$, use $X=x-\mu$, $Y=y$, $\nu =\mu -1$. The stable direction is the $X$ axis and the centre direction is the $Y$ axis. The extended centre manifold has $X=Y^2+O(\nu Y^2,Y^4)$ and reduced flow $\dot{Y}=-\nu Y-2Y^3+\cdots$, a supercritical pitchfork bifurcation in parameter $-\nu$. At $(0,1)$ near $\mu =-1$, use $X=x$, $Y=y-1$, $\nu =\mu +1$. The centre line is $Y=-X/2$, the stable line is $X=0$, and the reduced flow is $\dot{X}=\nu X-2X^2+\cdots$, a transcritical bifurcation. The branch $x=(\mu +1)/2$, $y=\sqrt{(1-\mu )/2}$ joins the two bifurcations and is stable for $-1<\mu <1$.