Whitney's Classification — Elementary Catastrophes in Control Parameter Space
Fold Catastrophe
Cusp Catastrophe
Bifurcation Set
Potential V(x) = x³/3 + ux
State Space: Equilibria
Control Parameters
Fold catastrophe (codimension 1): V(x) = x³/3 + ux. Equilibria at x² + u = 0. Stable branch: x < 0, u < 0. Fold point at u=0 (single root). Cusp catastrophe (codimension 2): V(x) = x⁴/4 + ax²/2 + bx. Equilibria: x³ + ax + b = 0. Bifurcation set in (a,b) plane: 4a³ + 27b² = 0 — the classic swallowtail curve. Inside the cusp, three equilibria coexist (hysteresis). Whitney's theorem guarantees these are the only stable singularities up to codimension 2.