Two Pendulums
A pendulum hung from the end of another pendulum follows fixed rules with no randomness. Start two copies a millionth of a radian apart and watch them part company.
Pendulum A (the start you chose) Pendulum B (nudged by the difference below)
How far apart they are
The vertical scale is logarithmic: each gridline is ten times the one below. The dashed line marks 0.1 metre. Each bob's arm is 1 metre, so the largest possible distance is 4 metres (top of the graph is 10 m).
What this is
The motion is the standard double pendulum: two equal rods of length 1 m with equal bobs of 1 kg, in gravity of 9.81 m/s². Its equations come from the Lagrangian and are integrated with a fourth-order Runge–Kutta step of 0.001 s. The energy error of pendulum A is shown so you can see how well the numbers keep to the rules.
The graph is a picture of chaos, not of randomness. Each run is fully determined by its start. The two bobs follow almost the same path for a while, then the gap roughly multiplies by ten every few seconds. Changing the nudge only moves that moment earlier or later. It does not make the motion any less predictable in principle, but it does make a long forecast fragile.