Nonlinear Pendulum — Separatrix & Phase Portrait

Phase portrait of the driven, damped nonlinear pendulum θ̈ + γθ̇ + sin(θ) = F·cos(ωt). The separatrix (energy E=2) divides libration from rotation. With driving, trajectories can become chaotic. Click canvas to add initial conditions. Poincaré section reveals the strange attractor.

Phase portrait (θ vs θ̇) — click to add trajectory

Separatrix at E = (θ̇)²/2 − cos(θ) = 1
Below: libration (bounded oscillations)
Above: rotation (full revolutions)

Chaos onset F ≈ 1.07 (γ=0.5, ω=2/3)