Two nearly identical starts, torn apart

Two points in a swirling flow begin so close together they are practically the same dot. Well below one exact dial-setting, both quickly fall onto the very same resting point — but creep close to that setting first, and the settling can take a long, chaotically wandering while. Cross the threshold itself, and however close together they began, they end up on opposite wings of the flow, with no way to tell they were ever almost the same point at all.

What you're seeing

Three equations, one dial. The point (x,y,z) drifts through space along a flow that always pulls it back toward the middle — but never lets it stop. ρ sets how hard the flow is driven. Below ρ=1, everything drains to a single still point at the origin, dead center. Above it, that center point turns into a launch pad instead of a landing spot, and two new calm points appear on either side; drag ρ up through the low teens and the flow spirals gently into whichever one it happens to start nearer. Keep dragging past about 24.7 and both of those calm points stop being calm — the flow can no longer settle anywhere, and instead loops forever between the two of them, first one wing then the other, on a shape that never repeats and never quite closes: the strange attractor.

That's when the two starting points — start A and start B, just 0.01 apart in a space where the whole shape spans nearly 50 units — stop behaving alike. Well below the threshold they head for the same landing spot and the tiny gap between them shrinks to nothing, usually within a few seconds — though drag ρ up to creep close to the threshold from below and that settling gets slower and messier, the two points wandering across the whole shape for a long while before finally locking together; a preview of the chaos to come, not a clean cutoff. Cross the threshold, and the same tiny gap grows instead, roughly doubling every fraction of a second, until within a few seconds the two points are on opposite wings of the butterfly with no correlation left at all — and this time they never do lock back together. Nothing random happened — the equations are exactly deterministic, run forward from an exact number — the flow itself simply amplifies any difference, however small, into a completely different future. Edward Lorenz found this by accident in 1963, rerunning a weather model from a rounded-off printout instead of the stored full-precision number, and watched the two "identical" re-runs diverge completely within a simulated month — the discovery that gave us the phrase "the butterfly effect."

The rule, exactly. ẋ = σ(y−x), ẏ = x(ρ−z)−y, ż = xy−βz integrated with fourth-order Runge–Kutta at a fixed step. σ=10 and β=8/3 are held at their classic textbook values; ρ is the one dial. The origin is a fixed point for every ρ, stable below ρ=1 and a saddle above it (one of its two eigenvalues crosses exactly zero at ρ=1, closed-form). For ρ>1 two more fixed points exist at (±√(β(ρ−1)), ±√(β(ρ−1)), ρ−1), stable until ρ_H = σ(σ+β+3) / (σ−β−1) ≈ 24.7368 — an exact closed form for σ=10, β=8/3, not a rounded estimate. Verified in node (improve/verify/50-lorenz.js): that closed form; the analytic eigenvalue sign-flip at ρ=1; a fixed-point-stability check by direct simulation at ρ=0.5, 10, and 28; and the largest Lyapunov exponent, measured the standard way (two trajectories, periodically rescaled back to a fixed tiny separation, log-growth averaged over time) — ≈+0.90 per time unit at ρ=28 (positive: nearby trajectories separate), matching the published value for the classic parameters. Negative control: the identical Lyapunov measurement at ρ=ρ_H−2 (just below the threshold) comes back negative — the very same two-trajectory test that finds chaos above the threshold finds ordinary, boring convergence just below it, and a control run at ρ=0.5 (below 1) converges even harder, losing over 15 decimal digits of separation in the time it takes the ρ=28 case to gain them. A direct long-run check in node, sweeping ρ from 22 up to 24.7, found exactly the pattern described above and no more: clean, fast convergence at ρ=22–23, and a genuinely long, non-monotonic chaotic wander at ρ=24 and above that hadn't fully resolved after 900 time units — real preturbulence near the bifurcation, not a bug in the copy or the code.

← the cabinet · Thought Toys — a cabinet of explorable explanations. Exhibit 50.