Two stable states, one unstable — slow parameter change causes sudden irreversible jumps (catastrophe)
Bifurcation Diagram + State
Potential V(x) = x⁴/4 − x²/2 − Ix
x = 0.00 | I = 0.00 | State: —
Normal form of cusp catastrophe: dx/dt = x − x³ + I. Fixed points satisfy x − x³ + I = 0.
Fold bifurcations at I = ±2/(3√3) ≈ ±0.385. Hysteresis: system jumps at different I depending on direction of sweep.
Green dot = current state. Solid curves = stable, dashed = unstable equilibrium.
Biological analog: genetic toggle switch (Gardner et al., Nature 2000).