Deterministic Nonperiodic Flow
Simple deterministic equations can be unpredictable forever.
Three tidy equations, run on a tiny 1960s computer, proved that the weather can be deterministic and still impossible to predict for long.
The big idea
Edward Lorenz was a meteorologist hunting for a simple toy model of convection — warm air rising, cool air sinking. He boiled it down to three numbers governed by three exact rules, with nothing random anywhere. Run the rules and the future is, in principle, completely fixed.
And yet the answer was unpredictable. If he started the model from two almost-identical states, they marched together for a while, then peeled apart until they had nothing in common. Tiny uncertainties in the start grow until they swamp everything — so even a perfect, deterministic model can only see a little way ahead.
How it happened
The discovery was an accident. In 1961 Lorenz wanted to re-run part of a simulation, and to save time he re-typed the numbers from a printout. The printout showed three decimals; the computer's memory held six. That difference — a few parts in a million — was enough that the re-run, after a while, produced a totally different 'weather'.
At first he suspected a broken vacuum tube. Then he realised the machine was right and his intuition was wrong: the system was exquisitely sensitive to its starting point. He stripped the model down to three equations to show the effect as plainly as possible, and published it in 1963 in a weather journal — where, for about ten years, almost no mathematician noticed.
Why it mattered
For three centuries, since Newton, science had assumed that exact laws meant a predictable world: know the present precisely and the future unrolls. Lorenz showed there is a catch. Many ordinary systems amplify the tiniest error so fast that long-range prediction is impossible in principle, not just in practice. 'Deterministic' and 'predictable' had quietly come apart.
This reshaped weather forecasting and gave us the popular name for the whole idea — the 'butterfly effect', from Lorenz's later image of a butterfly's wingbeat nudging a distant storm.
An everyday picture
Imagine two identical marbles released at the very top of a steep hill, a hair's breadth apart. At first they roll side by side. But the hill is studded with rocks, and each rock throws the marbles slightly differently. After enough bounces, one ends up in the pond and the other in the bushes — same launch, same hill, same rules, wildly different fate.
The atmosphere is that hillside with countless rocks. The butterfly's wingbeat is the hair's-breadth head start that decides, weeks later, which valley the marble lands in.
Where it sits
Lorenz turned a worry Henri Poincaré first glimpsed in the 1890s — that the solar system's three-body problem might be unpredictable — into something concrete and computable. His butterfly attractor opened the field of chaos, alongside Benoît Mandelbrot's fractals and Mitchell Feigenbaum's universal route to chaos.
It also drew a quiet boundary around the clockwork universe of Newton (1687) elsewhere in this Library: the laws stay exact, but the long-term forecast does not. The convection model Lorenz simplified leaned on the kind of mode-by-mode decomposition Joseph Fourier (1822) introduced.
Finite systems of deterministic ordinary nonlinear differential equations may be designed to represent forced dissipative hydrodynamic flow. Solutions of these equations can be identified with trajectories in phase space. For those systems with bounded solutions, it is found that nonperiodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can evolve into considerably different states. Systems with bounded solutions are shown to possess bounded numerical solutions.