Lorenz attractor

Can three simple deterministic equations be unpredictable — and how do we measure that?

▶ Run the simulationSee the measured result

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

How the lab tests it

Integrate the Lorenz system (σ=10, ρ=28, β=8/3) with RK4 alongside a near-identical Benettin twin started δ₀=1e-9 away; track the phase-space separation, renormalise it, and average its log-growth into the largest Lyapunov exponent λ.

What it checks

the textbook λ ≈ 0.9056 > 0 (exponential divergence ⇒ deterministic chaos), on the famous two-lobed butterfly attractor whose dissipative flow contracts phase volume at ∇·f = −(σ+1+β) = −13.667

Lorenz attractor calculator (Lyapunov spectrum, bifurcation ladder, predictability horizon)

In 1963 Edward Lorenz truncated Rayleigh–Bénard convection to three ordinary differential equations, restarted a run from a printout rounded to three decimals, and watched the new trajectory walk away from the old one. That is the butterfly effect, and the thing most descriptions of it never supply is a number. This page supplies the numbers. The headline one — the largest Lyapunov exponent λ₁ ≈ 0.9056 — is the one number here that has NO closed form and can only be measured, which is why this lab measured it rather than quoting it; everything AROUND it is exact algebra, and that is what a calculator can give you. The exact part starts with the divergence: ∇·f = −(σ+1+β) is a constant, the same at every point of phase space, so the three exponents must sum to −(σ+1+β) = −41/3 at the canonical parameters no matter what λ₁ turns out to be. Phase volume therefore contracts at a fixed rate while one direction stretches — which is the whole reason a zero-volume attractor with a positive exponent can exist at all. Feed this page a measured spectrum and it will tell you how close that sum lands, and read the Kaplan–Yorke dimension off it. The bifurcation ladder is exact too: below ρ = 1 the origin is the only attractor, at ρ = 1 it hands over in a pitchfork to the pair C±, and at ρ_H = σ(σ+β+3)/(σ−β−1) those two lose stability in a subcritical Hopf — 470/19 ≈ 24.7368 at σ = 10, β = 8/3, which this page computes as a ratio of two integers rather than quoting. One piece of arithmetic on this page is worth more than the rest, and it lives at the pitchfork. The origin's unstable eigenvalue is written in every textbook as λ₊ = (−(σ+1)+√((σ+1)²+4σ(ρ−1)))/2, and just above ρ = 1 that square root is barely larger than σ+1, so the subtraction throws away the answer's leading digits: at ρ = 1 + 10⁻¹², the textbook spelling is wrong by a relative 3.6e-4 and by ρ = 1 + 10⁻¹⁵ it is wrong by 12%, while the algebraically identical λ₊ = 2σ(ρ−1)/((σ+1)+√(…)) — the same expression cleared through its conjugate — stays correctly rounded all the way down. The onset of instability is precisely where anyone using this calculator would look, and it is precisely where the standard formula fails. The page prints both and counts where they part. What this page will NOT do: it will not compute λ₁ for you from σ, ρ and β, because no formula exists that does — it takes a spectrum you measured and prices it; it will not report a Hopf threshold when σ ≤ β+1, where the formula's denominator changes sign and the subcritical Hopf is simply not there to find; it will not give a doubling time for a non-positive exponent; and it will not revise a single number this lab recovered.