Normal form: ẋ = r + x²
At r=0: saddle-node bifurcation — stable and unstable fixed points collide and annihilate. For r<0: two equilibria. For r>0: none.
This is the fold catastrophe — the simplest catastrophe (Thom 1972). Adds cusp with extra parameter s: ẋ = s + rx + x³
Hysteresis: if you slowly change r, the system jumps at the fold — different paths give different outcomes.