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 at a counter-clockwise angle \theta from downward at time t 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 position varies sinusoidally

s = s_0 + A \sin \omega t

relative to the hanging rope, then its downward position

y = s + r \theta

and its acceleration

\ddot y = \ddot s + r \ddot \theta = -\omega^2 A \sin \omega t + r \ddot \theta.

The total downward force on the climber

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

implies upper rope tension

T = Mg-M \ddot y = M g + M\omega^2 A \sin \omega t-M r \ddot \theta.

If the axle and rope have negligible inertia, then the total torque about 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 + rM \omega^2 A \sin \omega t-Mr^2\ddot\theta-\gamma \dot\theta

rearranges to

(ml^2+Mr^2)\ddot \theta = -mgl\sin\theta + rMg + rM\omega^2 A \sin \omega t-\gamma \dot\theta.

For parameters that describe the animation’s chaotic motion,

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

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