The Lyapunov exponents λ₁ ≥ λ₂ ≥ ... ≥ λ_n quantify the average exponential divergence of nearby trajectories in each direction of phase space. A chaotic system has λ₁ > 0 — nearby orbits separate as eλ₁t. The Kaplan-Yorke dimension D_KY = j + (λ₁+...+λ_j)/|λ_{j+1}| (where j is the largest index with Σλᵢ ≥ 0) estimates the fractal dimension of the attractor. The Lorenz system has Lyapunov exponents ≈ (+0.906, 0, −14.57) and D_KY ≈ 2.062.
Lorenz attractor — trajectories diverging from nearby initial conditions
Running Lyapunov exponent λ₁(t) — should converge to +0.906