∂Systems · Chaos

Three numbers, one butterfly.

Edward Lorenz wrote down three differential equations in 1963 as a toy model of convection. He ran the integration overnight on a Royal McBee LGP-30. When he restarted from a printout, a tiny rounding error in the sixth decimal exploded into an entirely different weather. Determinism without predictability. The shape you are looking at is what those equations carve out in phase space.

RK1 integration · σ ρ β as sliders · two trails 0.0001 apart show divergence · prefers-reduced-motion respected

Three sliders, one attractor

The classical setting is sigma 10, rho 28, beta 8/3. Move rho down to 14 and the trajectory falls onto a fixed point. Push rho above 99 and chaos gives way to a stable period-three orbit. The boundaries between regimes are not smooth — chaos appears and disappears in narrow windows.

Why the shape matters

The Lorenz attractor was the first concrete example of a strange attractor in three dimensions. Its existence as a rigorous mathematical object was only proven by Warwick Tucker in 1999 with a computer-assisted proof. The shape carved out has Hausdorff dimension about 2.06: more than a surface, less than a solid.

1963 in Cambridge

Lorenz was running a simplified weather model on a Royal McBee LGP-30, slow even by 1963 standards. He stopped a run, wrote down the state to six decimals, and restarted from those numbers. The new run diverged from the original within hours of simulated time. The truncation from twelve to six decimals had not been an approximation but a small kick the system amplified exponentially. The Journal of the Atmospheric Sciences paper of that year is the seed of modern chaos theory.

The butterfly effect

In 1972 Lorenz gave a talk to the American Association for the Advancement of Science titled "Does the flap of a butterfly's wings in Brazil set off a tornado in Texas?" The title was suggested by a colleague who could not reach Lorenz before the program was printed. The image stuck. What it really says: sensitive dependence on initial conditions. Two states that look identical to any finite measurement separate at an exponential rate.

Strange and bounded

Every trajectory in this state-space, no matter where it starts, is pulled toward the same butterfly-shaped object. That is the attractor part. The strange part: the object has fractal structure. Look at it more closely and the wings reveal layers of wings. The Hausdorff dimension was estimated numerically at 2.062 by Grassberger and Procaccia in 1983.

Why weather forecasts have a limit

Real atmospheres are infinite-dimensional Lorenz systems, sensitive to tiny perturbations in temperature, humidity and wind. The doubling time of small errors is around two to three days. Numerical weather prediction can extend the useful range with better measurements and bigger ensembles, but not past two to three weeks. The chaos is in the equations, not in the computer.