Transcritical bifurcation

The normal form $\dot{x}=\mu x-x^2$ has equilibria $x=0$ and $x=\mu$ that cross and exchange stability at $\mu =0$. A generic constant perturbation separates the branches or replaces the crossing by saddle-node bifurcations, so the transcritical bifurcation is not structurally stable without a constraint preserving both branches.

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.