The exact pendulum equation θ̈ = −(g/L)sin(θ) has two qualitatively different motions. Below the separatrix: libration (oscillation). Above: rotation (spinning). The separatrix is the unstable manifold of the saddle at θ = π.
Energy H = ½θ̇² − (g/L)cos(θ). Separatrix: H = +g/L (just reaching top). Period diverges logarithmically near separatrix: T ~ ln(1/ε). With drive and damping: chaos near separatrix via heteroclinic tangle. The pendulum is integrable (elliptic integrals) without damping.