Lorenz Attractor

In 1963, Edward Lorenz was running a simplified weather model on a Royal McBee LGP-30 when he discovered that rounding an initial condition from 0.506127 to 0.506 produced a completely different forecast. The system of three ordinary differential equations — dx/dt = σ(y−x), dy/dt = x(ρ−z)−y, dz/dt = xy−βz — is fully deterministic. Given identical starting points, it produces identical trajectories. But make the starting points differ by any amount, no matter how small, and the trajectories diverge exponentially. This is sensitive dependence on initial conditions, colloquially known as the butterfly effect.

The shape you see above is a strange attractor: a set in phase space toward which all nearby trajectories converge, yet on which no two trajectories ever merge. It has fractal dimension — roughly 2.06 — meaning it is more than a surface but less than a volume. The system is dissipative: volumes in phase space contract. But the attractor itself has zero volume, an infinitely folded structure that trajectories approach forever without reaching. Every orbit is aperiodic. No state ever repeats exactly.

Enable multiple trajectories above to see the divergence directly. They start nearly identically, track each other for a while, then separate completely — one looping left while another loops right. This is deterministic chaos: the future is fixed by the present, but the present is never known with infinite precision. The system is not random. It is unpredictable. Those are different things.