If the equations are exact, why can’t we predict the outcome?

The meteorologist Edward Lorenz boiled rolling, heated air down to three quantities: $x$ is how fast it turns over, and $y$ and $z$ describe how far the temperature is from even. They change in time as

with the values he used: $\sigma=10$, $\beta=8/3$, $\rho=28$. There is nothing random in these equations: given a start, every later moment is fixed.

The paths on screen are computed with the fourth-order Runge–Kutta method and a time step of 0.005.

In this system, the distance between two neighbouring starts doubles on average about every 0.77 time units (the largest Lyapunov exponent is about 0.906). Exponential growth is fast: in the story the two points start $10^{-5}$ apart and are more than 1 apart after about 16 time units, after which they follow completely different paths.

In 1961 Lorenz re-ran part of a weather simulation and typed in a number rounded from six decimals to three; a few weeks into the run, the "weather" was unrecognisable. A 1972 talk of his asked whether the flap of a butterfly’s wings in Brazil could set off a tornado in Texas, and the "butterfly effect" got its name.

It is also one reason weather forecasts struggle to stay accurate beyond about two weeks: however precise the first measurements, errors grow like this.

Chaos is three things at once:

• Determined: nothing is random; the same start always gives the same result.
• Sensitive: tiny differences at the start grow exponentially.
• Bounded, never repeating: the path stays in a finite region but never retraces itself.

So no single point can be predicted for long, but the shape three thousand points end up spreading over is the same every time. It is called a strange attractor, and it is a fractal of dimension about 2.06, a little "more" than a surface.

$\rho$ stands for how hard the air is heated (σ and β stay fixed):

• $\rho<1$: too little heat; every point falls back to the origin and the air stays still.
• $1<\rho<24.06$: points settle at the centre of one wing, and the air rolls steadily one way (between 13.93 and 24.06 they may bounce around for a long time first).
• $24.06<\rho<24.74$: the butterfly and the two resting points coexist; where a point ends up depends on its start.
• $\rho>24.74$: both centres are unstable and only the butterfly is left.

The wing centres sit at $\left(\pm\sqrt{\beta(\rho-1)},\ \pm\sqrt{\beta(\rho-1)},\ \rho-1\right)$, about $(\pm8.49,\pm8.49,27)$ at $\rho=28$.