10,000 double pendulums start almost identical
Neighbours start 0.00000001 radians apart. For seven seconds they swing as one.
Δθ₀ = 0.00000001 radA double pendulum is one pendulum hanging from the end of another. The video runs 10,000 of them at once, all starting from the same angle, 2 radians, except that each one starts 0.00000001 radians away from its neighbour.
Seven quiet seconds
At first they move as one: the colours of all 10,000 overlap in a single swinging arm. The differences are far too small to see.
Then, at about 7 seconds, the arm frays and the pendulums scatter into every possible position.
Why the tiny difference wins
A double pendulum is chaotic. Two that start close together drift apart at a rate that keeps multiplying: the gap grows by roughly the same factor every fraction of a second. A gap of one hundred-millionth only needs a few dozen doublings to become as big as the swing itself.
The equations are exact and there is no randomness. The video is computed with the real equations of motion, 480 steps per second. The pendulums still become unpredictable, because predicting them would need the starting angle to infinite precision.
Start them all lower, at 0.5 radians, and they stay in step: small swings of a double pendulum are not chaotic.
