Three lines of code, one butterfly.
A toy model of the atmosphere that accidentally invented chaos theory. The Explorer integrates the equations live and lets you watch the trajectory refuse to repeat.
In 1963 a quiet meteorologist named Edward Lorenz wrote down three short equations to model a slice of the atmosphere. The equations are simple enough to fit on a postcard. The behaviour they produce is so wild it ended the dream of perfect long-range weather forecasts and started a whole new branch of science.
Start two simulations with positions that differ by 0.00001, about the width of a bacterium. For 20 seconds of simulated time the two paths trace the same butterfly wing. Then, almost without warning, one trajectory leaps to the other wing while the second stays put. After 30 seconds they are on opposite sides of the butterfly, totally uncorrelated.
This is the origin of the 'butterfly effect' and the reason your weather app stops being trustworthy after about two weeks. Lorenz didn't invent chaos — he revealed that it was hiding inside the cleanest equations we knew, and he gave us a vocabulary for living with it.
Drag the slider below to change ρ. Watch the shape collapse, breathe, and re-bloom into the butterfly.
Below 1: trajectory falls to the origin. Around 13.9: it settles to one of two fixed points. Past ≈24.7: the butterfly emerges.
Each equation describes how fast one quantity changes. x is the speed of the convection roll — warm air rising, cool air sinking. y is the horizontal temperature difference across the roll. z is the vertical departure from a linear temperature profile. The three pieces talk to each other: x pulls y, y pulls z, and z feeds back into y. The non-linear terms (xz and xy) are where chaos lives — without them the system would just be a damped oscillator.
dx/dt = σ (y − x) dy/dt = x (ρ − z) − y dz/dt = x y − β z
ρ (the Rayleigh number) measures how hard you are heating the fluid from below. Below ρ = 1 nothing interesting happens: any starting condition decays to zero — no convection, just heat conduction. Between 1 and about 24.74 you get steady convection, with the trajectory spiralling into one of two fixed points. Cross ρ ≈ 24.74 and the fixed points lose stability; at the canonical value ρ = 28 the trajectory can no longer settle anywhere — it is condemned to roam the butterfly forever.
No convection. Every trajectory dies into the origin.
Chaos. The trajectory never settles, never repeats.
Each wing is built around an unstable equilibrium — a point where the equations balance, but only fragilely. The trajectory spirals outward around one wing, gathering energy, until it suddenly slingshots across to the other wing and starts spiralling again. The number of loops on each wing before the jump is unpredictable. Sometimes 3, sometimes 17, never the same sequence twice — and that sequence is, in a precise sense, as random as a coin flip.
Each wing forms around one of the equilibria at (±√(β(ρ−1)), ±√(β(ρ−1)), ρ−1). They repel, and that repulsion is what keeps the trajectory in motion.
Two trajectories separated by ε grow apart roughly like ε·eλt, where λ ≈ 0.906 is the leading Lyapunov exponent. That sounds tame until you do the arithmetic: in about 16 simulated seconds the gap multiplies by a million. Any uncertainty in the initial state — measurement noise, rounding in a computer, a butterfly flapping in Brazil — is amplified at this rate. Weather forecasts decay because the real atmosphere has its own positive Lyapunov exponent, and no amount of better sensors can outrun an exponential.
Two trajectories. Same equations. Initial positions differ by a hundred-thousandth (10⁻⁵).
Every trajectory, no matter where it starts, is sucked onto the same butterfly-shaped object. That object is the attractor. It is not a closed curve (which would have dimension 1) and not a smooth surface (dimension 2) and not a solid blob (dimension 3). Its Hausdorff dimension is approximately 2.06: a surface with a fractal frill, infinitely thin in one direction but never quite flat. Zoom in on any patch and you find the same wafer-thin layered structure. This is what 'strange' means in 'strange attractor'.
Strictly between a surface (2) and a solid (3). Infinitely thin in one direction, fractally layered in another.
Lorenz's paper went almost unnoticed for a decade. Then James Gleick's 1987 book 'Chaos' put the butterfly on every coffee table, and the phrase 'butterfly effect' entered the language. The deeper lesson is harder than the slogan: deterministic equations can be utterly unpredictable, the world is full of systems like this (climate, ecosystems, economies, hearts), and prediction is not a question of computing harder — it has a fundamental horizon. Modern weather services run dozens of slightly perturbed forecasts at once for exactly this reason.
The Explorer integrates the three equations live, lets you change σ, ρ, β, and rotates the attractor in three dimensions. You can also launch two close trajectories at once and watch them diverge.
How and where this technique lives in the world today.
Topics in the same vein.
