Saddle-Node Bifurcation

The saddle-node bifurcation is the normal form ẋ = r + x². For r < 0: two fixed points (stable at −√|r|, unstable at +√|r|). At r = 0: they collide and annihilate. For r > 0: no fixed points — the trajectory passes through a ghost, slowing dramatically near where the fixed points once were. The passage time through the ghost scales as 1/√r, a key signature. Red curve = vector field; orange dot = current state.