Chaos Pendulum
A single pendulum, damped by friction and driven by a periodic force. At low drive amplitudes the motion is periodic. Increase the drive and the pendulum tumbles into chaos — a route to turbulence in the simplest possible oscillator.
What’s happening
The equation of motion
A pendulum with damping (friction coefficient q) and an external sinusoidal driving force:
d²θ/dt² + (1/q) dθ/dt + sin(θ) = F₀ cos(ωᵈ t)
Without the drive (F0 = 0), the pendulum oscillates and gradually settles to rest. With a weak drive, it locks into a periodic motion matching the drive frequency. But above a critical drive amplitude, the pendulum enters a chaotic regime where it tumbles over the top unpredictably.
Phase portrait
Plotting θ against dθ/dt reveals the geometry of the dynamics. Periodic motion traces a closed curve. Period-doubled motion traces two loops. Chaotic motion fills a bounded region densely — never exactly repeating.
Poincaré section
Sample the state (θ, dθ/dt) once per drive period. For periodic motion, the Poincaré section is a single point. For period-2, two points. For chaos, an intricate fractal structure appears. This is the stroboscopic portrait of the attractor.
The bifurcation diagram
Scan the drive amplitude F0 from low to high. At each value, let the system settle and record the Poincaré angles. You’ll see period-doubling cascades — the universal route to chaos discovered by Mitchell Feigenbaum. The ratio of successive bifurcation spacings approaches the Feigenbaum constant δ ≈ 4.669.
Why a single pendulum?
The double pendulum is chaotic because it has enough degrees of freedom. The driven damped pendulum is chaotic for a deeper reason: the external drive makes the system effectively three-dimensional (state = θ, dθ/dt, ωdt), satisfying the Poincaré–Bendixson condition for chaos. One degree of freedom plus a periodic force is the minimum recipe for deterministic chaos.