Equations
Fold: V(x) = x³ + ax
Critical: dV/dx = 3x² + a = 0
Bifurcation: a = 0
Cusp: V(x) = x⁴ + ax² + bx
Equilibria: 4x³ + 2ax + b = 0
Cusp point: a < 0 allows 3 roots
Thom (1972): Only 7 elementary
catastrophes in ≤4 control dims.
Codimension = # control params
The cusp catastrophe explains hysteresis: as control parameters change, the system can jump discontinuously — but the jump occurs at different points depending on direction of travel. Examples: buckling beams, heartbeat, opinion change.