Double Pendulum Chaos

Chaos Theory: The double pendulum is a canonical chaotic system. Equations of motion follow from the Lagrangian L = T - V, yielding coupled nonlinear ODEs for θ₁ and θ₂. Even arbitrarily small differences in initial angles grow exponentially — characterized by the positive Lyapunov exponent λ > 0. The right panel shows the angular separation between the first pendulum and each other: Δθ = |θ₁(t) - θ₁ᵢ(t)|. Initially parallel tracks diverge after just a few oscillations. The divergence time scales as ~(1/λ)·ln(ε₀/δ) where ε₀ is initial separation. Integrated with RK4 for accuracy.