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Set $g(x)=x^2+x^3/3$. Equilibria satisfy ${\mu }^2=g(x)$, and their scalar linear eigenvalue isIt vanishes only at $x=0$ or $x=-2$. Substitution in the equilibrium relation gives the three nonhyperbolic parameter-state pointsThe branches with $x<-2$ and $x>0$ are stable, while $-2<x<0$ is unstable. For $0<|\mu |<2/\sqrt{3}$ there are three equilibria; beyond the two outer saddle-node bifurcations there is just the positive stable one. At $\mu =0$ there is also the distinct stable equilibrium $x=-3$.

Near the crossing, use the smooth coordinate $y=x\sqrt{1+x/3}$, whose derivative is positive near zero. The equation becomes $\dot{y}=p(y)({\mu }^2-y^2)$ with $p>0$. A positive time rescaling removes $p$, and the parameter-dependent state shift $u=y-\mu$ givesThus the crossing at $\mu =0$ is a transcritical bifurcation, with parameter $-2\mu$. The two smooth branches $y=\pm \mu$ exchange dynamical stability; labeling the stable branch by $|\mu |$ instead would obscure that exchange. The parameter-dependent coordinate shift in a transcritical bifurcation explains why the original coordinate need not display a branch identically equal to zero.