Catastrophe Theory

Fold and cusp catastrophes — bifurcations, hysteresis, and sudden jumps

Fold Catastrophe
Cusp Catastrophe
Potential V(x)
Equilibria vs control a
-0.50
0.00
Fold catastrophe: V(x) = x³/3 + ax. As control a sweeps through 0, two equilibria (stable+unstable) coalesce and annihilate — a fold bifurcation. Cusp catastrophe: V(x) = x⁴/4 + bx²/2 + ax. The bifurcation set forms a cusp curve in (a,b) space; inside the cusp, two stable states coexist. Paths looping the cusp show hysteresis. Zeeman's catastrophe machine demonstrates these jumps physically.