Saddle-Node Bifurcation

Normal form: ẋ = r + x². For r < 0, two equilibria exist (stable + saddle). At r = 0, they collide and annihilate. For r > 0, no fixed points — the "ghost" slows trajectories where they used to be (bottleneck effect).

Phase portrait ẋ = f(x)

Bifurcation diagram

Parameter r: -0.50 Noise: 0.00

Show trajectories: Show ghost: Time span: 4

ẋ = r + x²

r = -0.50: two fixed points at x = ±√(-r) = ±0.71