The Lorenz System
In 1963 a weather scientist boiled the atmosphere down to three simple equations. The result never settles, never repeats, yet never escapes a strange butterfly shape. Drag to rotate it — and watch two near-identical starts go their separate ways.
The weather knob
Two starts
Order and disorder at once
The path never crosses itself and never repeats — but it also never leaves the two-winged shape. It's not random: random would fill the whole box.
Switch heating to Settles (ρ = 14) and the motion spirals quietly into a point — predictable weather. Flip back to Chaos (ρ = 28) and it never rests.
Same shape, different story
Both dots start almost exactly together and obey the exact same rules. For a while they move as one — then they jump to different wings and lose each other completely.
That's why a butterfly's wingbeat can, in principle, change the weather weeks later. The shape is predictable; where you are on it is not.
For teachers & grown-ups
These are the Lorenz equations — ẋ = σ(y−x), ẏ = x(ρ−z)−y, ż = xy−βz — with σ = 10, β = 8/3, integrated with 4th-order Runge–Kutta. ρ is the Rayleigh number (a stand-in for how hard the fluid is heated). For ρ < 1 the origin is stable; for 1 < ρ < ≈24.06 two stable points appear (the "Settles" preset spirals into one); above ≈24.74 the famous chaotic attractor takes over. The two trajectories differ only by the "tiny difference" in their starting x; their separation grows roughly exponentially (positive largest Lyapunov exponent ≈ 0.9 per time unit) until it saturates at the size of the attractor — the same sensitive-dependence story as the double pendulum and the logistic map, now in a continuous 3D flow. Worth emphasizing: the attractor is a fractal of zero volume that trajectories approach but never lie exactly on, and two nearby starts share the same attractor even as their individual paths diverge.