One Part In A Billion
Two identical pendulums, started a billionth of a radian apart. Watch how long identical lasts.
What it is
Two double pendulums, same equations, same integrator, same everything, except one starts with its upper arm nudged by 0.000000001 radians. About a hundredth of the width of an atom, measured at the end of the arm.
They trace the same path. Then they don’t.
How it works
The equations are the textbook ones, stepped with fourth-order Runge-Kutta at two milliseconds. That accuracy matters for an honest answer, because if the integrator were sloppy the divergence you see might be its error rather than the pendulum’s nature. So there’s a test demanding energy hold to within 1% across forty thousand steps, and it does.
Separation is measured in the full state, both angles and both angular velocities, and its growth is exponential, which means the slope of its logarithm against time is a Lyapunov exponent. It reads about 2 per second. Every half second the disagreement doubles.
What surprised me
I knew the divergence was exponential. What I hadn’t internalised was what that means for buying accuracy, until I measured it.
Time until the two pendulums disagree by an order-1 amount:
| starting nudge | time to diverge |
|---|---|
| 10⁻³ | 4.2 s |
| 10⁻⁶ | 6.7 s |
| 10⁻⁹ | 10.6 s |
| 10⁻¹² | 15.0 s |
A billion-fold improvement in how precisely you know the starting angle buys eleven extra seconds. Each thousand-fold gain in precision buys about three and a half.
And it isn’t even a disappointing result, it’s exactly right. Predicted extra time is
ln(ratio)/λ, so nine orders of magnitude at λ = 2 should buy ln(10⁹)/2 = 10.4 seconds.
Measured: 10.9. Theory and stopwatch agree to within five percent.
That’s the real content of chaos and it’s a much bleaker statement than “sensitive to initial conditions”. Prediction horizon grows like the logarithm of your precision. You can’t buy your way out with better instruments, you can only buy a few more seconds each time you spend a thousand times more.
What I would do next
Plot the divergence time against nudge size on a log axis and watch it come out as a straight line with slope 1/λ.