Parameter-dependent coordinate shift in a transcritical bifurcation ($\dot{y}={\mu }^2-y^2⟺\dot{u}=-2\mu u-u^2$)

A transcritical bifurcation need not already have its persistent branch at coordinate zero. The smooth parameter-dependent shift $u=y-\mu$ turns the displayed crossing of $y=\pm \mu$ into the standard form with parameter $-2\mu$. The crossing branches exchange dynamical stability. A mere nonsmooth relabeling of stable and unstable branches as $\pm |\mu |$ would conceal this smooth exchange.