Chaos and the integrator

Each pendulum in the fan starts a tiny angle to the right of its neighbour — by default a billionth of a radian. For a while they move as one. Then the fan opens: the distance between the first two grows exponentially, a straight line on the log chart, until they share nothing but the pivot.

The slope of that line is the largest Lyapunov exponent, estimated live with Benettin’s method. At λ ≈ 1.3 s⁻¹, every tenfold improvement in how precisely you know the start buys less than two extra seconds of prediction.

Explicit Euler steps along the current slope, which on a curved orbit always overshoots outward: the pendulum gains energy every step. RK4 is fourth-order accurate — at a 2 ms step its energy error stays below 10⁻⁷ through a chaotic half minute — but that error has a direction and keeps accumulating.

The implicit midpoint rule is only second order, yet it is symplectic: it preserves the geometry of Hamiltonian motion, so its energy error oscillates instead of growing. Try the coarse-step preset and compare the lines.

The habit transfers directly: pick the method for the property you cannot afford to lose — energy here, money in a ledger — and then check that property continuously instead of trusting the happy path.

This page’s physics is unit-tested against the analytic normal-mode frequency and against energy conservation, the same way reconciliation jobs verify that settlements and balances still add up.