The Double Pendulum
Two pendulums start in almost exactly the same spot. Watch what happens. Can you guess where each one will be in ten seconds?
Try it
Tip: set the tiny difference to 0 and the two pendulums stay glued together. Turn it up and they split apart.
A tiny difference becomes a big one
Both pendulums follow the exact same rules. The only difference is that the faint one started a tiny bit higher or lower than the bright one.
For a little while they move together — then suddenly they don't. After that, knowing where one is tells you almost nothing about the other.
Some things are hard to predict
This is called the butterfly effect: a tiny change at the start can lead to a completely different ending.
Weather works the same way. That's why forecasters can be right about tomorrow but not about three weeks from now — the tiny differences they can't measure grow too big.
For teachers & grown-ups
Both pendulums are simulated from the same equations of motion (solved with a 4th-order Runge–Kutta integrator). The "tiny difference" slider offsets the faint pendulum's starting angle by a fraction of a degree. Because the double pendulum is chaotic, that offset grows roughly exponentially — the visible moment when the two trajectories separate is the system's prediction horizon in action. Ask students to predict when they'll split, not just that they will; then try a smaller difference and see if the split happens later.