Cusp Catastrophe
V(x) = x⁴ + ax² + bx — control parameters (a,b) determine stable equilibria and sudden jumps
About: Catastrophe theory (René Thom 1972, Zeeman 1976) classifies qualitative changes in smooth dynamical systems. The cusp catastrophe V(x) = x⁴ + ax² + bx is the second simplest: equilibria occur where V'(x) = 4x³ + 2ax + b = 0. For a < 0, there can be up to three equilibria (two stable, one unstable). As parameters cross the cusp-shaped bifurcation set (27b² + 8a³ = 0), the system undergoes a fold bifurcation — one stable and one unstable equilibrium annihilate, causing a discontinuous jump. This models hysteresis in phase transitions, beam buckling, population crashes, and opinion tipping points. The canonical feature is that the path matters: the jump occurs at different parameter values depending on direction (hysteresis).