Thought Toys · Chaos & fractals · 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.) Counter-example, verified in node: at small lift the motion is regular, not chaotic — two arms started 1e-4 apart stay together (≈1× over 30 s), while at large lift they diverge by ~3×10⁵×.

Also in Chaos & fractals: The logistic map →

All 9 in Chaos & fractals
  1. 03The double pendulum — you are here
  2. 08The logistic map
  3. 20The Mandelbrot set
  4. 34Newton's fractal
  5. 50Two nearly identical starts, torn apart
  6. 74Push straight down, and it decides to lean
  7. 773n+1 always comes home. Swap in a 5, and it mostly doesn't.
  8. 90Pure randomness draws one exact shape, forever.
  9. 92Shred a picture completely. It puts itself back.

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