Structural stability of a bifurcation

A bifurcation is structurally stable within a specified class of systems when every sufficiently small allowed perturbation preserves its local qualitative bifurcation diagram up to smooth changes of state and parameter coordinates. A generic saddle-node bifurcation is structurally stable in one-parameter families, whereas a pitchfork bifurcation requires a symmetry and a transcritical bifurcation requires intersecting equilibrium branches, so unrestricted perturbations destroy either one.