Tiny Changes
Two pendulums start almost the same. The graph keeps score of how far apart they get. Will the line stay low… or shoot up?
Pick a swing
Try it
The line tells the story
For the simple swing, the line stays low — the two pendulums keep matching, so the future is easy to guess.
For the double swing, the line stays low for a moment… then suddenly shoots up to the top. After that, the two are completely different.
When the line shoots up, prediction ends
The flat part at the start is how long you can see into the future. Scientists call that the prediction horizon.
Try a smaller tiny difference. Does the line take longer to shoot up? That's exactly why better measurements let weather forecasters see a little further ahead.
For teachers & grown-ups
The graph plots the distance between the two pendulums' tips (as a percent of the largest possible separation) versus time. The double pendulum's curve rises roughly exponentially before saturating — the hallmark of chaos, and a direct, visual measurement of sensitive dependence on initial conditions. The simple pendulum's curve stays low and rises only slowly (a small, linear phase drift), because it isn't chaotic. The faint gray curve is your previous run, so students can compare simple vs double — or two different starting differences — on the same axes. Halving the "tiny difference" roughly delays the double pendulum's blow-up by a fixed amount of time: that constant relationship between starting precision and how far ahead you can predict is the Lyapunov time, without the vocabulary.