Bifurcation set

Summary

The bifurcation set of a smooth family of functions is a set of parameter values. It contains the values at which some function of the family has a degenerate critical point. It is the central object of catastrophe theory. Off this set the critical points move smoothly with the parameters, and across it they are created or destroyed. For the cusp family $x^4/4 + a x^2/2 + b x$ it is the cusp curve $4a^3 + 27b^2 = 0$ in the $(a, b)$-plane. For a family of potentials it describes where sudden jumps of the equilibrium can start.