Cusp Bifurcation: Codimension-2 Normal Form

Parameters

Unfolding r (horizontal) 0.00 Unfolding h (vertical bias) 0.00

ẋ = h + rx − x³

Fixed Points

Fixed points: —

Bistability: —

View

Show bifurcation set in (r,h) Show flow arrows

Theory

Cusp catastrophe (Thom 1972): The codimension-2 bifurcation where a fold bifurcation curve itself bifurcates. In parameter space (r,h), a cusp-shaped curve separates mono- from bistable regions.

On the cusp boundary: 4r³ + 27h² = 0

Inside the cusp: 3 fixed points (2 stable, 1 unstable). Outside: 1 stable fixed point. Crossing the fold triggers a sudden jump — catastrophe.