In 1961 Edward Lorenz was running a weather simulation and, to save time,
restarted it from the middle using a printout rounded to three decimal places
instead of six. The result was completely different. A difference of 0.000127
had, after months of simulated time, produced an entirely divergent forecast.
The butterfly effect was born.
Deterministic but unpredictable.
The Lorenz equations have no randomness. Given identical starting conditions,
they produce identical results — always. But the system is exquisitely sensitive
to initial conditions: two points separated by 10⁻¹⁵ will diverge exponentially.
In practice, we never know starting conditions to infinite precision.
Determinism and predictability are not the same thing.
The attractor itself.
Despite the chaos of individual trajectories, the system never leaves a bounded region.
It spirals around two lobes forever, never repeating, never escaping.
The shape it traces — the attractor — has fractal dimension ~2.06.
It is more than a surface but less than a solid. This is the "strange" in strange attractor.
The twin trajectories (red and blue) start at points separated by 0.001 —
imperceptible. Watch how long it takes before they split. That time scale is the
Lyapunov time: the horizon of predictability. For the atmosphere, it is
roughly two weeks — which is why weather forecasting beyond that window is fundamentally impossible,
not just technically difficult.
The other attractors — Rössler, Halvorsen, Dadras, Thomas — each arise from
different three-variable systems. The Rössler is simpler: one lobe, one spiral, with occasional
large excursions. Thomas is fully symmetric in all three axes. Each one is a different face of
the same underlying truth: simple rules, iterated, produce infinite complexity.
drag to rotate · adjust parameters to deform the attractor · watch twin trajectories diverge