Nonlinear Pendulum Phase Portrait

θ̈ = −(g/L)sin θ − bθ̇ + A·cos(ωt)
θ (rad)
0.00
θ̇ (rad/s)
0.00
Energy E/mgl
0.00
Click phase plane to set initial condition.
Red curve = separatrix (energy = 2)
Blue = libration (oscillation)
Orange = rotation (over-the-top)
The nonlinear pendulum θ̈ = −(g/L)sin θ has two qualitatively different motions: libration (oscillation around θ=0) and rotation (continuous spinning). The separatrix E = 2mgL divides these regimes. Adding damping creates a stable spiral at the origin. Adding periodic driving (Duffing-like) can produce chaos — the Poincaré map reveals fractal structure when A is large enough.