Hysteresis and snap-through transitions — the system has two stable states; which one it occupies depends on its history
dx/dt = I + x^n/(1+x^n) − γx
With cooperative self-activation (n>2), the system has two stable fixed points separated by an unstable one. Moving the parameter I up and down traces different paths — this is hysteresis.
• Biological: genetic toggle switches (Gardner 2000), cell fate decisions
• Chemical: autocatalytic reactions with degradation
• The fold bifurcation (saddle-node) creates and destroys the fixed points