The most famous shape in chaos theory, spinning live in your browser. A single point traces the Lorenz equations forever, looping around two lobes into the iconic butterfly — never repeating, never crossing itself. Drag to rotate it in 3-D, turn the ρ dial, then split off a twin trajectory a millionth apart and watch them diverge. It's the 3-D companion to the bifurcation diagram and the double pendulum — the same butterfly effect, drawn in space.
A millionth of a difference, and the future is unrecognisable — yet always the same butterfly. Edward Lorenz found this in 1963 when rounding a weather-model number from 0.506127 to 0.506 sent the forecast somewhere completely different. Run the twin here — a second trajectory started just 1e-3 away from the first, under the identical rule dx=σ(y−x), dy=x(ρ−z)−y, dz=xy−βz with σ=10, ρ=28, β=8/3. The two paths track together, then peel apart onto opposite wings. Neither ever leaves the attractor; neither ever repeats. Determinism does not mean predictability.
How It Works
Three equations, integrated step by step, tracing a shape that never repeats.
1
Integrate the equations
The state is a point (x, y, z). Three coupled rules — dx=σ(y−x), dy=x(ρ−z)−y, dz=xy−βz — say how it moves each instant. We advance it with fourth-order Runge–Kutta, a small, accurate time step repeated many times per frame, so the path stays true even as it stretches and folds.
2
Watch the strange attractor form
Wherever it starts, the point is quickly pulled onto the same butterfly-shaped surface and orbits one lobe, then the other, in an order that never repeats. That surface is the strange attractor — bounded, fractal, and chaotic. Drag to rotate it, or let it spin; lengthen the trail to see the full shape at once.
3
Split the twin, see the butterfly effect
Turn on the butterfly twin and a second point starts a mere 1e-3 away. The separation readout climbs exponentially as the two diverge onto different wings — sensitive dependence on initial conditions. Try Rössler and Aizawa for other strange attractors born from the same idea.
What is the Lorenz attractor?
The Lorenz attractor is the set of trajectories traced out by the Lorenz equations, a simplified model of atmospheric convection published by meteorologist Edward Lorenz in 1963. The three equations dx=σ(y−x), dy=x(ρ−z)−y, dz=xy−βz, with the classic constants σ=10, ρ=28 and β=8/3, produce a path that loops endlessly around two spirals without ever repeating, forming the famous butterfly-shaped figure in three dimensions.
What is a strange attractor?
An attractor is the shape a system settles onto in the long run — a point, a loop, or a surface that nearby trajectories are drawn toward. A strange attractor is one with fractal structure on which the motion is chaotic: trajectories stay confined to a bounded region yet never repeat and are exquisitely sensitive to their starting point. The Lorenz attractor was the first strange attractor to be widely studied, which is why it became the emblem of chaos theory.
What is the butterfly effect?
The butterfly effect is sensitive dependence on initial conditions: in a chaotic system, an arbitrarily tiny change in the starting state grows exponentially until the outcome is completely different. Edward Lorenz found this while rounding a weather-model input from 0.506127 to 0.506, and the name comes from his talk asking whether a butterfly's wings in Brazil could set off a tornado in Texas. Turn on the twin trajectory here — started just 1e-3 away — and watch the two paths, identical at first, drift onto opposite wings of the attractor.
Why does the trajectory never repeat or cross itself?
The Lorenz system is deterministic, so its future is fixed by its current position — which means if the path ever crossed itself it would be forced into a repeating loop. Because the motion is chaotic and bounded on a strange attractor of fractional dimension, it must thread endlessly through the same region without ever landing on a point it has visited before. In three dimensions the curve comes close to itself again and again but never truly intersects.