The exact pendulum equation θ̈ = −(g/L)sin θ − γθ̇ has two qualitatively different behaviors: libration (oscillation around equilibrium) and rotation (continuous spinning). These are separated by the homoclinic orbit — the separatrix — which connects the unstable fixed point to itself in phase space (θ = ±π, θ̇ = 0).
Click on the phase portrait to launch a trajectory