Where's the maths? A pendulum is trigonometry set in motion: the height of a bob is L cos θ and the sideways pull that swings it back is proportional to sin θ, so the whole machine is built out of the sine and cosine of Sec 2. Turning that into a rule for how θ changes from one instant to the next needs calculus — a differential equation, which is simply a statement about a rate of change. A single pendulum obeys its rule tamely and repeats itself forever, which is why it can run a clock. Hang a second rod from the bottom of the first and the same honest trigonometry produces something new: motion that never repeats. The branch of mathematics that studies this is dynamical systems, better known as chaos theory. Nothing on this screen is random — the computer solves identical equations for both pendulums to about fifteen digits — yet a difference of one hundredth of a degree at the start grows into a difference of a whole swing in under twenty seconds. That is exactly why a weather forecast is useful for a week and useless for a month.

The state of a double pendulum is just four numbers: the two angles θ₁, θ₂ and the two turning speeds ω₁, ω₂. Newton's laws turn into two equations giving the angular accelerations in terms of those four numbers and nothing else — no luck, no dice, no hidden input. The computer takes tiny steps of 1/1200 of a second and uses the Runge–Kutta method to follow them.

Start two copies a hair apart and the gap between them does not creep up — it multiplies:

• λ is the Lyapunov exponent. Here λ ≈ 0.4 per second, so the gap multiplies by e ≈ 2.7 roughly every 2½ seconds — about ten times bigger every 6 seconds.
• That is why the graph plots log₁₀ of the gap: taking logarithms turns multiplying into adding, so exponential growth becomes a straight line. The fitted slope, times ln 10, is λ.
• Because a starting gap of 0.01° ≈ 10^−4.8 rad has to climb only about 5 decades to reach a whole radian, a hundredfold better measurement buys you just 2 more decades — about 12 extra seconds. Precision loses to exponentials.
• The energy readout is the honesty check. Gravity and the rods do no net work, so the total energy must stay fixed; ours drifts by less than a millionth of a percent in a minute, so the wild motion is physics, not numerical rubbish.
• Switch to Single pendulum and the same two starts stay locked together: one rod is regular, and for small swings its period is T = 2π√(L/g) — no mass, no amplitude, just the length. (Release it from 90° and the readouts show the real period running about 18 % long: "small angle" really does mean small.) One extra joint is the whole difference between predictable and unpredictable.