The exact period of a nonlinear pendulum is T = 4√(L/g) · K(sin(θ₀/2)), where K is the complete elliptic integral of the first kind. As the amplitude θ₀ → π (approaching the unstable equilibrium), K(k)→∞ and the period diverges. The phase portrait shows the separatrix between libration and rotation.
Period: — s | k = sin(θ₀/2) = —