The butterfly effect is a number and the lab measured it

Can three simple deterministic equations be unpredictable — and how do we measure that? Lorenz's 1963 discovery says nearby atmospheric states diverge exponentially, so forecasting has a horizon; the quantitative content is the largest Lyapunov exponent λ₁, a number with NO closed form that can only be measured.

Units: nats per unit sim time (largest Lyapunov exponent; Sprott 2003 canonical spectrum (0.9056, 0, −14.5723))

▶ Run this simulationRead how it works

The finding

The butterfly effect is a number and the lab measured it: the Lorenz attractor's full Lyapunov spectrum recovered as (0.90587±0.00036, −0.0003, −14.5722) vs the literature (0.9056, 0, −14.5723) — λ₁ to 2.9e-4 relative, 0.74 SE, never fed — with the exact dissipation identity Σλ=−(σ+1+β) landing at 4.5e-7, the SAME equations provably regular at ρ=10 (analytic eigenvalues to 5.8e-4) and ρ=350 (limit cycle, λ₁=0), and the derisk EXECUTING the shipped module: default screen pinned strict (λ bit-exact, trail + clouds sha256, 193-char HUD ===) and the executed RK4's origin tangent map equal to the stability polynomial R(hλ) at machine epsilon, Richardsoning to the continuum eigenvalues (−11±√1201)/2, −8/3 and the exact −41/3

Method

Integrate the Lorenz system (σ=10, ρ=28, β=8/3) with RK4 at dt=0.005 together with its EXACT tangent flow — three orthonormal frame vectors driven by the Jacobian evaluated at the RK4 stage states (variational RK4) — and Gram–Schmidt-renormalise the frame every 0.5 time units (Benettin et al. 1980); the accumulated log norms are the full Lyapunov spectrum. 24 seeded random initial conditions, 100 time units of transient discarded, 5000 time units measured per seed. λ₁ is recovered a second, independent way — the live module's finite-difference twin at δ₀=1e-9 — and the two methods must agree per-seed. Controls at ρ=10 (fixed-point regime, measured exponents scored against the ANALYTIC eigenvalue real parts of the Jacobian at C± from the characteristic cubic) and ρ=350 (stable limit cycle, λ₁=0). The known spectrum (Sprott 2003) is loaded only to score. ?world=lorenz.

The law it recovers

sensitive dependence on initial conditions: infinitesimal phase-space separations grow as e^(λ₁t) with λ₁ ≈ 0.9056 — a doubling time of ln2/λ₁ ≈ 0.77 time units, the quantitative butterfly effect

Measurements, controls & cross-checks

Recovered uncertainty

0.00036

Recovered se distance

0.74

Spectrum

Lambda1

0.905867

Lambda2

-0.000312

Lambda3

-14.57223

Lambda2 exact

0

Lambda2 note

λ₂ = 0 is EXACT for any bounded non-fixed-point attractor of an autonomous flow (the flow direction neither grows nor shrinks) — recovered at 3.1e-4 absolute, never imposed

Lambda3 rel offset

5.8000e-6

Dissipation identity

Sum measured

-13.66666

Sum exact

-13.666667

Rel offset

4.5400e-7

Worst seed rel

4.5500e-7

Note

λ₁+λ₂+λ₃ = ∇·f = −(σ+1+β) = −41/3 exactly (the flow contracts phase volume at a CONSTANT rate — why a zero-volume attractor exists at all). The analytic number is never used by the recovery; the QR log-volume lands on it to 4.5e-7, limited only by RK4's O(dt⁴) determinant error. The identity also holds in the periodic regime (ρ=350: 8.7e-7).

Kaplan yorke

Measured

2.06214

Known

2.06215

Abs offset

7.4000e-6

Note

D_KY = 2 + (λ̂₁+λ̂₂)/|λ̂₃|: the attractor is a fractal barely thicker than a surface — the geometry of chaos read off the measured spectrum

Controls

Fixed point rho10

Measured

• -0.596076
• -0.595419
• -12.47517

Analytic re eigenvalues

• -0.595497
• -0.595497
• -12.475672

Worst abs offset

0.000579

Note

same equations, ρ inside the stable window (< 470/19, and below the ρ≈13.926 homoclinic explosion): trajectories spiral into C± and the Benettin machinery must reproduce closed-form linear algebra — the roots of λ³+(σ+β+1)λ²+β(σ+ρ)λ+2σβ(ρ−1)=0. It does, to 5.8e-4. Chaos is NOT a consequence of nonlinearity.

Limit cycle rho350

Lambda1

0.000271

Lambda1 exact

0

Lambda2

-0.731

Note

past the last chaotic regime (ρ ≳ 313, Sparrow 1982) the attractor is a stable symmetric limit cycle: λ₁ = 0 (flow direction), no positive exponent — order returns at extreme driving. The same instrument found λ₁ < 0, = 0 and > 0 across the three regimes.

Numerics

Dt halving shift

0.000751

Gs interval invariance

1.1900e-14

Twin vs tangent worst

0.00024

Note

GS-interval invariance at 1.2e-14 because the same trajectory is re-bookkept; the finite-difference twin (the module's method) independently confirms the tangent-flow λ₁ per-seed to 2.4e-4

Module cross check

Module replica lambda

0.8958

Oracle lambda1

0.905867

Abs offset

0.0101

Note

exact headless replica of the live module's measurement (twin from (0.1,0,0)+δ₀x̂, dt=0.01, renorm 0.5, settle 2.0, T=3000): the on-screen λ carries a ~1% low bias from its short 2-time-unit settle (launch transient inside the average) — quantified, inside the 0.02 method-agreement tolerance

Honest module certificate

Executed

src/modules/LorenzModule.ts (sha256-pinned, 32 exact strip pairs + 21/7/16 asserted bulk counts, new Function + Babylon/DOM recorder stubs; init(), fixedUpdate() and render() all run headless)

Screen pinned strict

the default ?world=lorenz screen driven 10800 engine ticks at fl(1/120): λ = 0.68796792110918747 bit-exact, Σln(d/δ₀) and 176 renorms exact, 38 lobe hops, trail ring + sign buffers and both rendered lobe point clouds (2140+1460 pts) sha256-pinned, the full 193-char HUD and the 2800-char divergence-chart SVG strict-===

Accumulator disclosed

the module's float engine-tick accumulator delivers 8999 RK4 steps where the rational 10800·(1/120)/0.01 schedule says 9000 (and 299999/300000 over the 3000 s run) — the fl(1/120) one-step deficit pinned as exact integers, not idealized

Screen bias disclosed

the 90 s on-screen λ = 0.688 vs known 0.9056 is a FINITE-TIME fluctuation of the 88 s Benettin window (expected sd ≈ 0.1 by 1/√T from the T=5000 per-seed sd), not a bias: the same executed module at 3000 s lands at 0.89616392014710444 (bit-exact pin), within 9.7e-3 of the oracle λ̂₁ and 3.8e-4 of the gate-I replica — the replica can no longer silently diverge from the source it mirrors

Screen oracle closed

displayed 'theory 0.9056' === known_spectrum.lambda1 === reference known_value === the executed LAMBDA_THEORY static (all strict), displayed λ/∇·f/lobe-hops are the correctly-rounded executed pins, and gate A independently ties λ̂₁ to the same known within 0.74 SE

Executed integrator calibrated

the origin is a BIT-EXACT fixed point of the shipped _rk4 (1000 steps); a central-difference tangent probe (δ=1e-5 — a renormalized twin would freeze its rounding at a fixed point) shows the executed one-step map equals the RK4 stability polynomial R(hλ) at machine epsilon (2.2e-16) for all three exact Jacobian eigenvalues λ± = (−11±√1201)/2, λz = −8/3, with the z-block decoupling EXACTLY (0.0); Richardson (h, h/2) recovers the continuum eigenvalues to 5.9e-8 / 3.8e-6 / 8.5e-12 and their sum −41/3 to 3.9e-6 — none fed to module or recovery; the instrument's finite-dt bias is closed-form and disclosed: ln(R(hλ₊))/h − λ₊ = −1.75e-5 at h=0.01, leading order −λ₊⁵h⁴/120 (ratio 0.906)

Zero module edits

true

Seeds

24

What it reduces to

Lorenz's deterministic nonperiodicity (E. N. Lorenz, J. Atmos. Sci. 20, 130 (1963)): a 3-mode truncation of Rayleigh–Bénard convection whose bounded solutions never repeat and separate exponentially — the first strange attractor. It VALIDATES, not derives: the lab assumes the ODEs and shows (i) the largest Lyapunov exponent — a number with no closed form, only measurable — equals the canonical literature value 0.9056 (Sprott 2003) to 2.9e-4 with λ never fed; (ii) two EXACT structural laws emerge unforced: λ₂ = 0 (flow direction, Haken 1983) and Σλ = −(σ+1+β) = −41/3 (constant divergence, Lorenz 1963 §4) to 4.5e-7; (iii) the Kaplan–Yorke dimension 2.06215 follows from the measured spectrum; and (iv) chaos is a parameter REGIME, not a property of nonlinearity — at ρ=10 the same equations settle to fixed points whose measured exponents match the analytic eigenvalues of the linearisation at C± (characteristic cubic, closed form) to 5.8e-4, and at ρ=350 they lock onto a limit cycle with λ₁ = 0. The recovery is the Benettin tangent-frame + Gram–Schmidt method (Meccanica 15, 9 (1980)), cross-checked by an independent finite-difference twin. It does NOT claim anything about the PDE convection problem the truncation came from, and the fractal dimension is inferred via Kaplan–Yorke, not box-counting. The lab's first measured strange-attractor spectrum: the discrete-chaos Feigenbaum world (?world=logistic) quantified the ROUTE to chaos; this world quantifies chaos itself.

Confidence & reproduction

Confidence

high

Validation

derisk-pass

Re-run the check

npm run derisk -- lorenz (scripts/lorenz-derisk.mjs)

Oracle

scripts/oracles/lorenz.reference.json

Sources

E. N. Lorenz, 'Deterministic Nonperiodic Flow', J. Atmos. Sci. 20, 130–141 (1963). G. Benettin, L. Galgani, A. Giorgilli, J.-M. Strelcyn, Meccanica 15, 9–30 (1980). A. Wolf, J. B. Swift, H. L. Swinney, J. A. Vastano, 'Determining Lyapunov exponents from a time series', Physica D 16, 285–317 (1985). J. C. Sprott, Chaos and Time-Series Analysis (Oxford, 2003), Appendix A. J. Kaplan & J. Yorke, Lecture Notes in Math. 730, 204 (1979). C. Sparrow, The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors (Springer, 1982).