Thought Toys · Exhibit 03

The double pendulum

Hang one pendulum off another and the swinging turns wild. There is no luck in here and no hidden noise — every arm obeys the same exact equation. Yet release a fan of them from almost exactly the same place and, after a few honest seconds together, they fly apart and never agree again.

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The pack
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A spray of nearly identical double pendulums that begin as one and scatter over time.

gentle · stays in stepflung high · chaos
near-identicalvisibly apart

What you're seeing

One pendulum is predictable — it ticks back and forth like a clock. Bolt a second one to its end and the whole contraption stops being tidy: it whirls, flips, and stalls in patterns that never quite repeat. Nothing random is happening. Each arm just keeps trading energy with the other under plain Newtonian gravity.

The exhibit drops a whole flock of these at once, each lifted from almost the same angle — a fraction of a degree apart, close enough that they launch as a single stripe. For a moment they swing as one. Then a microscopic disagreement at one joint gets amplified by the next swing, and the next, until the flock blows open into a spray of colors all going their own way. That blow-up is sensitive dependence on initial conditions — the technical heart of chaos, and the reason long-range weather forecasts give out.

Now drag how high you lift them down low. Released from a gentle angle, the same flock stays welded together far longer — small swings are nearly orderly. Chaos isn't in the machine; it's switched on by how hard you push it.

The rule, exactly. Two equal rods (length 1, mass 1) under gravity g = 9.81. With both angles θ measured from straight down, the standard equations of motion are integrated with classic fourth-order Runge–Kutta at a 0.004 s step. Each pendulum in the fan starts from rest, its angle offset by a sliver from its neighbour. The model is exact and deterministic: rerun from identical numbers and you get an identical path. The scatter you see comes only from those slivers. (Checked offline: total energy holds to ~0.004% over a minute, and a 0.0001-radian nudge grows about e-fold every 0.7 s at full lift — a positive Lyapunov exponent.)

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