= Connecting pitchfork and transcritical branches in a quadratic-product flow
{title2=$\dot x=x(\mu-x+y^2),\quad\dot y=y(1-x-y^2)$}
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 $-1<\mu<1$.
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