Normal form ẋ = r + x² · fixed points · potential landscape · phase portraits
A saddle-node bifurcation is the most fundamental way fixed points are born and destroyed. The normal form is ẋ = r + x². For r < 0, there are two fixed points x* = ±√(−r) (one stable, one unstable). At the bifurcation point r = 0, they coalesce. For r > 0, both fixed points disappear — the system has no equilibrium and trajectories diverge.
The potential is V(x) = −rx − x³/3. The bifurcation diagram shows stable branches as solid lines and unstable as dashed. Also available: pitchfork (ẋ = rx − x³, symmetry-breaking) and transcritical (ẋ = rx − x², exchange of stability).