On Sensitive Dependence

or, how a butterfly's wing changes the weather
FIELD: nonlinear dynamics
MTHD: numerical (RK4, k=4)

In 1961, the meteorologist Edward Lorenz restarted a weather simulation from the middle, rounding 0.506127 to 0.506. The forecast diverged so dramatically that he asked himself whether the flap of a butterfly's wing in Brazil could set off a tornado in Texas. The answer, in a precise sense, is yes. What follows are three windows onto the phenomenon: a continuous flow in three dimensions, a mechanical system you can almost touch, and a humble parabola that bifurcates its way into chaos. Drag the sliders. Compare trajectories. Watch the separation explode.

FIG. 1 — STATE SPACE, PROJECTED & ROTATING
FIG. 2 — DIVERGENCE OF TRAJECTORIES (LOG SCALE) — THE BUTTERFLY EFFECT, MADE QUANTITATIVE

Parameters

σ — Prandtl number10.00
ρ — Rayleigh number28.00
β — geometric factor2.667
ε — initial separation0.0001
trail length2000.00

Presets

§3.1The equations

Lorenz's system arose from a drastic simplification of atmospheric convection. Three variables describe a fluid layer heated from below: x for the rate of convective motion, y for the horizontal temperature variation, and z for the vertical. Their evolution is governed by:

dx/dt = σ(y − x)
dy/dt = x(ρ − z) − y
dz/dt = xy − βz

At σ=10, ρ=28, β=8/3, the classical values, the system has no stable equilibrium but is bounded. Trajectories never repeat, yet they are forever drawn into a butterfly-shaped set of zero volume — a strange attractor. Look at FIG. 2: starting two trajectories within 10⁻⁴ of each other, their separation grows roughly exponentially until it saturates at the size of the attractor itself. The slope of that line, on average, is the system's largest Lyapunov exponent, λ ≈ 0.906. Predictability dies at the rate λ.

§3.2What changes with the parameters?

Drag ρ down to ~13 and the butterfly collapses into a stable spiral — two fixed points, no chaos. Push ρ past 99.65 and you find narrow windows of periodic orbits embedded in the chaos. σ controls how fast x is dragged toward y; β sets the geometry of the attractor's wings. The hard part isn't writing the equations down. The hard part is that knowing the equations exactly still doesn't let you predict the future.

References
  1. Lorenz, E. N. (1963). Deterministic Nonperiodic Flow. Journal of the Atmospheric Sciences, 20(2), 130–141.
  2. Li, T.-Y. & Yorke, J. A. (1975). Period Three Implies Chaos. American Mathematical Monthly, 82(10), 985–992.
  3. May, R. M. (1976). Simple mathematical models with very complicated dynamics. Nature, 261, 459–467.
  4. Feigenbaum, M. J. (1978). Quantitative universality for a class of nonlinear transformations. Journal of Statistical Physics, 19(1), 25–52.
  5. Poincaré, H. (1908). Science et Méthode. Flammarion.
RK4 INTEGRATION · dt = 0.005 · 4 STEPS/FRAME
« the present determines the future, but the approximate present does not approximately determine the future » — Poincaré