Double pendulum

A dozen pendulums are released at the same instant. Their initial angles differ by 0.1°. For the first two seconds they are indistinguishable; after that they take completely different paths.

The equations are deterministic: four numbers (θ₁, ω₁, θ₂, ω₂) fix everything. But neighbouring initial conditions separate as e^(λt), where λ is the largest Lyapunov exponent. Under the default settings λ ≈ 3–5 s⁻¹.

Prediction fails not because the model is wrong, but because initial conditions can never be measured exactly. Choosing a smaller ε only delays the split; it does not remove it.