= Steady-state bifurcation
{title2=$\det D_xf(x_0,\mu_0)=0$}
A change in local branches or stability of <equilibrium points of a dynamical system> as a parameter varies. A zero Jacobian eigenvalue is necessary for the equilibrium-branch bifurcations of a smooth finite-dimensional system: without it, the <implicit function theorem> gives a unique nearby equilibrium branch. It is not sufficient by itself; nonzero coefficients in the reduced <normal form of a dynamical system> establish the bifurcation type. A Hopf crossing has nonzero imaginary eigenvalues and is a different local bifurcation.
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