The butterfly equations
Why even a perfect rule can make the future impossible to predict.
What you are looking at
The dots trace a looping path shaped like two wings. They go around one wing a few times, jump to the other, and keep going, never quite repeating.
The idea
In 1961 a weather scientist named Edward Lorenz reran a computer weather model, but typed in a number rounded to fewer digits. The new run began the same but drifted further and further from the old one until the weather was completely different.
His model was only three numbers that change by simple rules. No dice, nothing random. Yet a tiny difference at the start grows until the two results have nothing to do with each other. This is why weather forecasts get unreliable after a week or two, no matter how good the computer.
When you draw the path of the three numbers, you get a shape like a butterfly. The path goes around one wing, then the other, and never crosses itself and never repeats. The picture is a map of how this kind of chaos is organized.
Where else you see it
- Weather forecasts that lose accuracy after several days.
- A pinball bouncing off bumpers: a tiny change in the start makes a big change in the end.
- The "butterfly effect" idea.
Go deeper: the proper names and the math
The Lorenz equations are dx/dt = σ(y − x), dy/dt = x(ρ − z) − y, dz/dt = xy − βz, with the classic values σ = 10, ρ = 28, β = 8/3. They come from a heavily simplified model of a fluid heated from below (convection).
Nearby paths separate roughly like eλt with a largest Lyapunov exponent λ ≈ 0.9 (per unit time), which limits how long a prediction is useful. The set the paths settle onto is a strange attractor: it has zero volume and a fractal dimension of about 2.06.