Tipping Points — Fold Catastrophe & Bifurcation

Fold catastrophe: ẋ = μ + x − x³. Stable fixed points (solid) and unstable (dashed) annihilate at saddle-node bifurcations μ = ±2/3·√3. Hysteresis: system jumps at forward tipping point but does NOT return until reverse tipping point — irreversibility is the hallmark of catastrophes. Real examples: lake eutrophication, ice-albedo feedback, ecosystem collapses, financial crashes.