The Butterfly Effect Lab
Two "weather worlds" start almost exactly the same — they differ by a tiny amount you choose. Your job: measure how long it takes before the two weathers become completely different. Then change the starting difference and see if you can predict what happens.
Set up the experiment
Smaller = the two worlds start closer together. How does that change the result?
Right now
| Starting difference | Time until the worlds split | Gap doubled every |
|---|---|---|
| Run the experiment to record your first measurement… | ||
Hunt for the straight line
On the Normal graph the gap looks flat, then suddenly explodes. Switch to Stretched (log) and that explosion turns into a neat straight line — the gap was doubling at a steady rate the whole time, even while it looked tiny.
That steady doubling is the fingerprint of chaos. Steady doubling adds up to a sudden surprise.
Does starting closer help?
Make the starting difference 10× smaller and run again. Does the "time until split" get 10× longer? Check your table — it doesn't. You only buy a small, fixed amount of extra time.
That's the catch with weather: even a perfect 10× better measurement only pushes the forecast a few days further out, never to infinity.
For teachers & grown-ups
Both worlds are the Lorenz system (σ = 10, ρ = 28, β = 8/3, RK4), started on the attractor and offset by the chosen difference in x. Their separation grows like d(t) ≈ d₀·e^{λt} until it saturates at the attractor's size — so on a logarithmic axis it's a straight line whose slope is the largest Lyapunov exponent (λ ≈ 0.9 per time unit here). The "gap doubled every…" figure is ln 2 / λ, fit from the clean exponential band of each run. The key quantitative lesson lives in the data table: because growth is exponential, the time to reach a fixed separation scales with the logarithm of the starting difference. Cutting d₀ by 10× adds only ln(10)/λ ≈ 2.6 time units — a constant, not a multiplier. That logarithmic wall is exactly why halving weather-measurement error extends useful forecasts by days, not weeks.