The Lorenz Attractor: How a Butterfly Changes the Weather

Three ODEs, one butterfly

Edward Lorenz, an MIT meteorologist, simplified a twelve-equation atmospheric convection model down to three coupled ordinary differential equations, where x is the rate of convective overturning, y the temperature difference between rising and sinking currents, and z the deviation of the vertical temperature profile from linear:

dx/dt = σ(y − x)
dy/dt = x(ρ − z) − y
dz/dt = xy − βz
classic chaotic values: σ = 10, ρ = 28, β = 8/3

With these parameters, the trajectory never repeats and never escapes to infinity — it spirals around one lobe, unpredictably crosses to the other, spirals there, and crosses back, tracing the two-winged "butterfly" shape that has become chaos theory's most recognisable icon. Lorenz published his findings in 1963; in a 1972 lecture he posed the famous question — does a butterfly's wings in Brazil set off a tornado in Texas?

live demo · a strange attractor tracing its butterfly wings● LIVE

Lyapunov exponents and the predictability horizon

How fast do two nearby trajectories diverge? The answer is the largest Lyapunov exponent λ₁ ≈ +0.906 for the classic parameters — a small initial error δ₀ grows as δ(t) ≈ δ₀·e^(λ₁t). Because this is exponential, the predictability horizon T_p ≈ (1/λ₁)·ln(Δ/δ₀) grows only logarithmically with measurement precision: halving the initial error buys only about 0.76 extra time units. This is why global weather models with 10⁹ degrees of freedom still lose forecast skill after roughly 7–10 days — the fault lies with chaos itself, not with imperfect computing power.

λ₁ ≈ +0.906   λ₂ ≈ 0.000   λ₃ ≈ −14.572
D_KY = 2 + (λ₁+λ₂)/|λ₃| ≈ 2.062   ← Kaplan-Yorke fractal dimension

The attractor's fractal dimension is barely above 2 — thin as a surface, but with infinite fine layered structure. Operational forecasters respond to this fundamental limit with ensemble forecasting: running dozens of simulations from slightly perturbed initial conditions and reading uncertainty from how far apart they spread.

Frequently asked questions

What are the three Lorenz equations?

dx/dt = σ(y−x), dy/dt = x(ρ−z)−y, dz/dt = xy−βz. With the classic parameters σ=10, ρ=28, β=8/3, the system never settles into a periodic orbit and never diverges — the trajectory is trapped forever on a fractal "strange attractor" shaped like a butterfly.

What is the butterfly effect, precisely?

It is the statement that nearby trajectories in a chaotic system separate exponentially fast, at a rate set by the largest Lyapunov exponent (λ₁ ≈ 0.906 for the classic Lorenz parameters). Even a perturbation smaller than an atom's diameter eventually makes long-term prediction impossible, which is why Lorenz asked whether a butterfly's wings in Brazil could set off a tornado in Texas.

Why can't weather be forecast more than about 10 days ahead?

The atmosphere is a chaotic system with a positive Lyapunov exponent, so any measurement error in the initial conditions grows exponentially. Because the predictability horizon only grows logarithmically with measurement precision, making instruments a thousand times more accurate buys just a few extra days of skill, not months.

Try it live

Everything above runs in your browser — open Lorenz Attractor — 3D Butterfly Effect, launch two nearby trajectories and watch them diverge exponentially in real time. Nothing is installed, nothing is uploaded.

▶ Open Lorenz Attractor simulation