Steady-state bifurcation ($\det D_{x}f(x_0,{\mu }_0)=0$)

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.