Simulation · as seen in our chaos-fan reel
Chaos fan
100 double pendulums start almost exactly together, each 0.000001 rad apart. For a few seconds they swing as one. Then they don't.
Time0.00 s
Spread now0
Still together100/100
First to break—
All split—
≈ 1 µm at the tip between neighbours
Start pose
What you're seeing
Each pendulum has two 1 m arms with a 1 kg weight on each, and there's no friction. They don't touch each other: each one is its own simulation, drawn on top of the others. Pendulum #1 (red) starts in the reel's pose. Each next one has its lower arm a tiny bit further round, 0.000001 rad by default, which is about a thousandth of a millimetre at the tip.
- The tiny differences grow exponentially. For a while the fan looks like a single pendulum. Then it smears into a rainbow, and finally every pendulum goes its own way. The rules are fully deterministic, yet the outcome depends on differences far too small to measure. That's chaos.
- Spread now is the largest distance between any two tips. Still together counts the pendulums whose tip has never been more than 10 cm from #1's tip (#1 counts itself). With the reel's pose and the defaults, violet #100 breaks away first at 7.93 s and the last one (#2) leaves at 12.92 s, as in the reel.
- Make the difference 1,000 times smaller and it only buys a few more seconds: 10⁻⁹ rad splits at 19.0 s instead of 12.9 s. That's what exponential growth does.
- RK4 with a fixed step of 1/2400 s, the same arithmetic as the reel. Energy drifts by less than 0.000005% over 24 s.