Saddle-Node (Fold) Bifurcation

Normal form: ẋ = r + x²
At r = 0, two fixed points (x*= ±√−r for r<0) collide and annihilate — the fold catastrophe. For r>0 no fixed points exist; trajectories slow dramatically near where x*≈0 would be, creating a ghost — a bottleneck causing critical slowing down. Passage time through the ghost scales as τ ~ r⁻¹/². The bifurcation diagram shows a characteristic parabolic fold: the upper branch is unstable (saddle), lower branch stable (node), merging at the cusp.