Lorenz attractor
The Lorenz attractor is the butterfly-shaped set that the Lorenz equations trace in three dimensions. A point circles one of two wings, then switches to the other at moments nobody can predict.
Equations
Three variables and two nonlinear terms, xz and xy. Lorenz derived them from a model of convection: x is the rate of the rolling motion, y and z describe how temperature varies across and up the cell. The system is symmetric under a half turn about the z axis, (x, y, z) to (−x, −y, z), which is why the two wings are mirror copies.
The divergence of the flow is −(σ + 1 + β), about −13.67 everywhere. Any small volume of starting points shrinks by a factor of e13.67 each time unit, which is how the whole of space collapses onto a thin, layered set.
History
Edward Lorenz, a meteorologist at MIT, published the system in 1963 in Deterministic Nonperiodic Flow, in the Journal of the Atmospheric Sciences. It is a drastic simplification of a model of fluid heated from below: warm fluid rises, cools and sinks in rolls.
The discovery came from a rounding. In 1961, restarting a weather simulation partway through, Lorenz typed in 0.506 where the computer had stored 0.506127. The new run tracked the old one briefly and then bore no resemblance to it. His 1972 talk, Predictability: Does the Flap of a Butterfly's Wings in Brazil Set Off a Tornado in Texas?, gave the effect its popular name. The attractor's shape looking like a butterfly is a coincidence that helped the name stick.
What to look for
Watch the red head as it comes back toward the middle, between the wings. On each pass it either loops the same wing again or crosses over, and the number of loops before a switch follows no pattern.
Turn it edge on and the wings flatten into what look like two sheets. The fractal layering is in their thickness: zoom in and each sheet is many sheets.
The app centres the view on z = 27, the height of the two fixed points the wings circle.
The butterfly effect, live
Two points start 0.00001 apart in x and follow the same equations with the same step. For a while their trails lie on top of each other. Then the gap grows by a factor of e about every 1.1 time units, until the two are on different wings at different times and knowing one tells you nothing about the other.
Separation against time, log scale. The dashed line is growth at the largest Lyapunov exponent, about 0.906 per time unit.
The numbers
| (0, 0, 0) | fixed point at the origin, unstable for ρ > 1 |
| (±8.49, ±8.49, 27) | the two fixed points the wings circle, (±√(β(ρ − 1)), ±√(β(ρ − 1)), ρ − 1) |
| ρ ≈ 24.74 | where those two fixed points lose stability, for σ = 10 and β = 8/3 |
| λ ≈ 0.906 | largest Lyapunov exponent: nearby paths separate by e every 1.1 time units |
| ≈ 2.06 | fractal dimension of the attractor |
Questions
- Why is it called the butterfly effect?
- Because of Lorenz's 1972 talk, "Predictability: Does the Flap of a Butterfly's Wings in Brazil Set Off a Tornado in Texas?" The phrase describes sensitive dependence on initial conditions: a tiny change in the starting state grows until the outcome is completely different. The butterfly shape of the attractor came first and is a coincidence.
- What parameters make the Lorenz system chaotic?
- The classic values are σ = 10, ρ = 28 and β = 8/3, which Chaos Studies uses. With σ and β fixed at those values, the two fixed points lose stability at ρ ≈ 24.74, and above that the familiar chaotic attractor is what you see.
- Is the Lorenz attractor a fractal?
- Yes. Its dimension is about 2.06: more than a surface and much less than a solid. Cut through a wing and you find layers, and each layer is made of more layers.