Lyapunov exponent

The Lyapunov exponent λ quantifies sensitive dependence on initial conditions: for two trajectories starting at nearly identical phase-space points, the separation grows as |δx(t)| ≈ |δx(0)| · e^(λt) on average. A chaotic system has at least one positive Lyapunov exponent.

For the atmosphere, λ ≈ 1/(2 weeks) — after two weeks, initial-condition uncertainty has grown by a factor of e, and deterministic weather prediction is essentially useless. For the solar system, λ ≈ 1/(5 million years), which is why planetary positions cannot be predicted beyond a few tens of millions of years into the past or future.

Defined by Aleksandr Lyapunov in his 1892 doctoral thesis on the stability of motion; extended to the multiplicative ergodic theorem by Oseledets in 1968.