λ = lim (1/N) Σ ln|f'(x)| for the logistic map xₙ₊₁ = rx(1−x)
λ = — | regime: —
The Lyapunov exponent measures the average exponential divergence of nearby trajectories.
For the logistic map, λ = lim (1/N) Σ ln|r(1−2xₙ)|. λ<0 → stable periodic orbit;
λ>0 → chaos. The onset of chaos occurs near r≈3.57 (period-doubling accumulation point).
Top panel shows the orbit; bottom shows the running average of ln|f'| converging to λ.