cusp catastrophe

René Thom, 1972. A system in equilibrium. Two control parameters, a and b. One state variable, x. As you move through control space, the equilibrium shifts — smoothly, until suddenly it cannot. Then it jumps.

x³ + ax + b = 0 · the cusp surface

control space

drag the point through (a, b) space. the cusp is the boundary.

current position a = 0.00, b = 0.00

x = −2 state variable x x = +2

catastrophe manifold

the surface x³ + ax + b = 0 in (a, b, x) space. your position, marked.

legend

stable equilibrium

unstable branch

bifurcation set (cusp)

current state

Inside the cusp: three equilibria exist — two stable, one unstable. Outside: only one. The catastrophe happens when you cross the fold curve while inside, and the stable branch you're on vanishes.

potential function

V(x) = ¼x⁴ + ½ax² + bx

equilibria are minima/maxima of V

equilibrium condition

dV/dx = x³ + ax + b = 0

solved for x given (a, b)

bifurcation set

4a³ + 27b² = 0

the cusp curve: where roots collide

Thom called them elementary catastrophes — not disasters, but forms. The seven ways a smooth system can suddenly reorganize itself. The cusp is the second simplest, and the most familiar: opinion tipping, heartbeat rhythm, the moment a bridge buckles.

What you are watching is not chaos. It is topology. The surface is perfectly smooth — the jump is forced by the geometry of what cannot exist in between.