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.

Green mass sinusoidally climbs up and down brown rope torquing blue pendulum into chaotic motion.