The logistic map showed chaos in one discrete variable. But it took a continuous system of three coupled ODEs — a stripped-down weather model — to reveal chaos's true geometric home: the strange attractor. In 1963 the meteorologist Edward Lorenz, truncating the equations of atmospheric convection to just three variables, stumbled on an object that had no right to exist. His system was

with the classic parameters \sigma = 10, \beta = 8/3, \rho = 28. Trajectories neither settle to a fixed point nor close into a cycle. They are drawn onto a beautiful, infinitely layered surface — the Lorenz attractor — and wander over it forever without ever repeating.

The butterfly

Below is the (x,z) projection of a single Lorenz trajectory: the famous butterfly. The orbit loops around one wing, then — with no warning and no pattern — flips to the other, circling an unpredictable number of times on each before switching again. Slide the parameter \rho to watch the attractor being born: for small \rho everything decays to the origin; past \rho \approx 24.7 the two wings light up and chaos sets in.

Two facts make this object strange. First, it is an attractor: nearby trajectories are pulled onto it and volumes in phase space shrink to zero (the flow is dissipative). Second, it is a fractal — not a surface, not a curve, but something in between, of non-integer dimension roughly 2.06. An attractor that is simultaneously attracting and fractal, on which motion is chaotic, is what we mean by a strange attractor.

Sensitive dependence: the butterfly effect

The defining property of chaos is sensitive dependence on initial conditions. Start two trajectories a whisker apart and they stay together only briefly before separating exponentially. Below, two Lorenz trajectories begin 0.001 apart in x; their x(t) traces overlap perfectly for a while, then peel apart and become completely uncorrelated:

Quantitatively, an infinitesimal separation \delta_0 grows like

where \lambda is the largest Lyapunov exponent. A positive Lyapunov exponent is the crisp, quantitative signature of chaos: it says errors double roughly every \ln 2 / \lambda \approx 0.77 time units. This is why Lorenz declared that a butterfly flapping its wings in Brazil might set off a tornado in Texas — and why weather is unpredictable beyond a couple of weeks no matter how good our measurements or computers. The doubling of tiny errors is unbeatable.

Worked example: how long until forecasts fail?

Suppose we know the state to one part in a million, \delta_0 = 10^{-6}, and a forecast is useless once the error reaches order one, \delta = 1. With \lambda = 0.906, when does that happen?

Step 1 — set up the growth law. \delta = \delta_0 e^{\lambda t}, so

Step 2 — evaluate. \ln(10^6) = 6\ln 10 \approx 13.8, so t \approx 13.8 / 0.906 \approx 15.2 time units.

Step 3 — read the moral. Improving our initial knowledge by a factor of ten — a huge experimental effort — buys only \ln 10 / \lambda \approx 2.5 extra time units of skilful forecast. The exponential is a wall: you cannot out-measure chaos, you can only chip at the edges of the horizon.