How it works: Each pendulum solves the same Lagrangian equations (θ₁, θ₂ for upper and lower bobs) via RK4, but member k starts with θ₁ offset by k·Δθ. Left canvas: all pendulums rendered simultaneously — colours range from violet (reference) to yellow (most different). As long as pendulums stay together, they look like one. When chaos kicks in they fan out. Right canvas: log-scale plot of |θ₁ − θ₁_ref| vs time; an exponential divergence (straight line on log plot) means chaos with positive Lyapunov exponent λ₁. The displayed λ is estimated by linear regression on the log-separation in the divergence window.

Left panel: Ensemble fan coloured violet→yellow.   Right panel: log₁₀ separation vs time — exponential growth → chaos.