Lorenz Attractor

The system

These are Lorenz's equations of 1963, written with his y and z exchanged, because the app is y-up and the textbook writes them z-up. σ = 10 sets the first line, β = 8/3 damps the vertical, and ρ = 28 sits in the third. The only nonlinear terms are the two products, x·z in the second line and x·y in the third, and they carry the whole figure.

Why it never settles

The field has three rest points. The origin is a saddle: it pulls in along two directions at rates −22.8 and −2.67, and throws out along a third at +11.8. The other two are the centres of the wings, at x = z = √(8/3 · 27) = 6√2 ≈ 8.49 and height y = 27. There the eigenvalues are −13.9 and 0.094 ± 10.2i. A seed circles a wing every 2π/10.2 ≈ 0.62 seconds, and creeps outward by about half a percent per turn.

Nothing can rest, so a seed winds out until the saddle in the middle catches it and hands it to the other wing, at intervals no rule predicts. That is why the threads, which begin almost on top of one another, are strewn over the whole figure within seconds. Neighbouring paths separate by a factor of e about every 1.1 seconds, and ten-fold in under 3.

History

Edward Lorenz was running a numerical weather model and found that restarting it from a rounded printout gave a wholly different forecast. He boiled convection down to these three equations for Deterministic Nonperiodic Flow in 1963, and the pictured object became the first strange attractor most people ever saw.

Try

• Top in the deck: the mask from above, the two lobes wrapped around their centres.
• Change the 28 in the third line to 15. Below ρ = 24.74 the wing centres are stable, and every seed spirals quietly into one of them, at x = z = 6.11, height 14.
• Set Seeds to 1: a single path draws the whole attractor on its own, given time.
• Drop Seed spread to its lowest and watch how long the pack holds together before it scatters.

Read more

• Lorenz system
• Edward Norton Lorenz
• Butterfly effect
• Attractor