Forecast horizon ($n_{\mathrm{forecast}}\approx {\lambda }^{-1}\log (\Delta /{\epsilon }_0)$)

A forecast horizon is the duration over which a prediction meets a specified error tolerance. If an initial uncertainty ${\epsilon }_0$ grows approximately as ${\epsilon }_0{e}^{\lambda n}$ with positive Lyapunov exponent $\lambda$, then solving for when it reaches a tolerance $\Delta$ gives the displayed estimate. Here $\log$ is the natural logarithm and $\lambda$ is measured per step of the discrete dynamical system; for a flow map, use the exponent per unit time instead. The estimate assumes the growth approximation remains valid up to the chosen tolerance and excludes additional model errors.

Reducing ${\epsilon }_0$ by a factor ten buys approximately $(\log 10)/\lambda$ more steps, rather than multiplying the forecast duration by ten. For typical logistic map orbits at $r=4$, $\lambda =\log 2$, so one extra decimal digit buys roughly $3.32$ steps in this approximation. Better measurements help, but their benefit is logarithmic. A forecast of a probability distribution or a stable long-term average asks a different question from predicting each individual state.