Three simple equations whose weather never repeats. Feed a state in, the flow winds forever around two wings and never closes a loop; two starts a hair apart come out worlds apart — the butterfly effect. The blue team builds and defends the attractor; the red team tries to break it. This is a model — a three-mode truncation of convection, not the weather.
source Lorenz, Deterministic Nonperiodic Flow (1963), J. Atmos. Sci. 20:130–141 · DOI 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2 — journals.ametsoc.org. Rendered, not quoted.
Convection stripped to three modes (Saltzman 1962; Lorenz 1963):
ẋ = σ(y−x)
ẏ = x(ρ−z)−y
ż = xy−βz
Classic: σ=10, ρ=28, β=8/3. Fixed points: the origin, plus (for ρ>1) two centres C± at (±√(β(ρ−1)), same, ρ−1) — here ±8.485, z=27, checked live.
The flow contracts volume everywhere: divergence ∇·f = −(σ+1+β) = −13.667 < 0, so the attractor has zero volume. Live state of trajectory A:
| x | y | z | |A−B| |
|---|---|---|---|
| — | — | — | — |
1963: a three-variable convection toy whose bounded aperiodic flow founded chaos theory. Deterministic yet unpredictable — the butterfly effect got its name from the wing-shaped attractor and from Lorenz's talk title.
Its bounded-but-never-repeating orbit is the same signature as the-double-pendulum: a low-dimensional, fully determined system that no forecast can pin down for long. Each sphere is the next one's initial condition.
The blue team's live check: re-integrate the classic system and confirm divergence<0, the twin-1e−8 divergence trend, and that ρ<1 decays to 0. If red tampers, this badge catches it.
The machine takes a 3-vector state x, y, z and three parameters:
| symbol | meaning | value |
|---|---|---|
| σ | Prandtl number | 10 |
| ρ | Rayleigh ratio | 28 |
| β | geometry | 8/3 |
Two starts are fed at once: A = (1,1,1) and its twin B = (1+1e−8,1,1). Integrated with RK4, fixed step dt=0.006. Named assumptions: well-posed truncation, no noise — a model, not a weather forecast.
Drag ρ below 1 to watch the wings collapse into the origin — the same collapse the red team weaponises. Everything is integrated live; no frame is looked up.
The machine's proven output for the classic parameters: the flow is BOUNDED (stays forever in a finite ball), APERIODIC (z never settles to a point or a cycle), VOLUME-CONTRACTING (∇·f<0 ⇒ zero-volume attractor), and shows SENSITIVE DEPENDENCE (a 1e−8 twin diverges by >5 orders of magnitude — a positive Lyapunov exponent). For ρ<1 the origin is the only fixed point and is globally stable.
The blue witness (left) confirms these live; the red team (right) tries to make them false.
And "chaos" is a claim about a model with named assumptions (well-mixed modes, constant parameters, no noise). The full rigorous proof that this attractor is genuinely strange stayed open for 39 years — only settled by Tucker's computer-assisted proof (2002). Determinism does not buy predictability.
"A butterfly's wing causes a tornado." Cut. The effect is sensitive dependence — a tiny error in the initial state grows exponentially. No wing causes anything; forecasts just lose skill.
"Chaos means random." Cut. The system is fully deterministic — no randomness anywhere. Same state in, same state out, forever. Only initial-condition error is amplified.
"Lorenz set out to find chaos." Kept, corrected. He found it by accident in 1961, restarting a run from a rounded printout (0.506 for 0.506127) and getting a wildly different forecast.
The red team's move: quietly drop ρ below 1. Now the origin is the only fixed point and it is globally stable — every trajectory decays to 0, the wings vanish, the chaos is gone.
Push ρ<1 and both the bounded-aperiodic test and the sensitive-dependence test fail — the witness (window 7) re-integrates, finds the flow collapsing to a point, and turns red. Nothing is faked; the attack is real and it is caught.