Butterfly Effect

Lorenz (1963) discovered that a deterministic 3-variable ODE — dx/dt = σ(y−x), dy/dt = x(ρ−z)−y, dz/dt = xy−βz — produces aperiodic trajectories with sensitive dependence on initial conditions. Two trajectories starting distance Δ₀ apart diverge exponentially: Δ(t) ≈ Δ₀ e^λt, where λ ≈ 0.9 is the maximal Lyapunov exponent. Despite chaos, the attractor has a fractal dimension ≈ 2.06 — structure persists at all scales. The "butterfly" shape arises because trajectories orbit two unstable foci before switching wings unpredictably.