= Parameter-dependent coordinate shift in a transcritical bifurcation
{title2=$\dot y=\mu^2-y^2\quad\Longleftrightarrow\quad\dot u=-2\mu u-u^2$}
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.
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