Nonlinear Pendulum: Exact Phase Portrait
Exact phase space dynamics with separatrix, libration orbits, and driven chaos
Controls
Phase Space
Pendulum
Poincaré
Parameters
Damping (b)
0.00
Drive amplitude (A)
0.00
Drive frequency (ω)
0.67
Initial Conditions
θ₀ (angle)
0.50
ω₀ (velocity)
0.00
State
θ (rad)
—
ω (rad/s)
—
Energy E/mgl
—
Orbit type
—
Reset
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Physics
Click phase space to set initial condition. Red curve is the separatrix (E = 2mgl).
θ̈ + b·θ̇ + sin(θ) = A·cos(ωt)
E = ½ω² - cos(θ)
E < 1: libration (oscillation). E > 1: rotation. Separatrix at E = 1 (exactly). Driven+damped → chaos!