Catastrophe Theory: Fold & Cusp

Thom's catastrophe theory classifies generic discontinuities (catastrophes) of smooth maps. The fold (codimension 1) and cusp (codimension 2) are the two simplest. The cusp describes hysteresis: a system can be in two stable states with a region of bistability bounded by the cusp bifurcation curve.

Cusp Bifurcation Set

Potential Well

Hysteresis Path

Parameters

Control a: 0.0 Control b: -0.5 Sweep speed: 1.0

Cusp catastrophe:
V(x)=¼x⁴+½ax²+bx

Equilibria:
∂V/∂x = x³+ax+b = 0

Bifurcation set:
4a³+27b²=0
(discriminant = 0)

Applications:
• Buckling beams
• Heartbeat rhythm
• Opinion dynamics
• Phase transitions