Thom's classification: fold and cusp catastrophes, bifurcation surfaces, and discontinuous jumps
Parameters
Fold catastrophe: V(x) = x³/3 + ax
Critical set: x² + a = 0 (fold curve: a = -x²)
At a < 0: two equilibria (min + max). At a > 0: no real equilibria. The jump at a = 0 is the fold.
State Trajectory
Thom's theorem (1972): any smooth 1-parameter family of generic gradient systems near a singularity is locally equivalent to one of 7 elementary catastrophes. The fold and cusp are the simplest two — codimension 1 and 2 respectively.