The normal form ẋ = r + x² creates or destroys equilibria at r = 0. A fundamental mechanism of sudden state transitions.
Normal form: ẋ = r + x² + bx³. For r < 0: two fixed points x* = ±√(−r), stable (lower) and unstable (upper). At r = 0 they collide in a saddle-node (fold). For r > 0: no real fixed points — trajectories escape to infinity. This underlies sudden transitions in climate, ecology, neurons, and engineering. The bifurcation diagram (right) shows stable (solid) and unstable (dashed) branches.