Simplest Chaos

The motion of one of the simplest dynamical systems, a torqued, damped, nonlinear pendulum, can be infinitely complicated.

Consider a simple pendulum of length l and mass m rigidly connected to an axle of radius r wrapped by a rope that hangs down one side with a mass M climbing up and down it, as in the attached animation.

If the climber’s height

x = x_0  + \frac{a}{\omega^2} \sin \omega t,

varies sinusoidally (relative to the axle), then its acceleration

\ddot x = – a \sin \omega t,

also varies sinusoidally, so the total force on the climber

M\ddot x = \sum_\text{down} f = Mg-T

implies upper rope tension

T = Mg + ma \sin \omega t,

where 0 < a < g. If the axle and rope have negligible inertia, then the total torque on the axle

m l^2 \ddot \theta = \sum_\text{CCW}\tau = – mgl \sin\theta + rT – \gamma \dot\theta,

where \gamma is the axle viscosity. The full motion equation

m l^2 \ddot \theta = – mgl \sin\theta + rMg + rma \sin \omega t – \gamma \dot\theta

reduces to

\ddot \theta = – \sin\theta + 0.7155 + 0.4 \sin 0.25 t – 0.75 \dot\theta

for parameters that describe the animation’s chaotic motion.