Two nearly identical trajectories diverge exponentially — the geometric meaning of chaos
Phase Space (x-z projection)
Separation |Δr(t)| — log scale
Lyapunov exponent λ ≈ 0.906 (theoretical)
Lorenz (1963): dx/dt = σ(y−x), dy/dt = x(ρ−z)−y, dz/dt = xy−βz with β=8/3. Two trajectories starting at (x₀,y₀,z₀) and (x₀+ε,y₀,z₀) diverge exponentially. The straight line on the log-separation plot gives the Lyapunov exponent λ ≈ 0.906. After divergence saturates (attractor diameter), trajectories wander independently — long-term prediction is impossible.