Lyapunov time

The Lyapunov time of a chaotic system is the inverse of its largest positive Lyapunov exponent, τ = 1/λ_max. It represents the characteristic time over which the system "forgets" its initial conditions: after one Lyapunov time, a perturbation of size ε has grown by a factor of e, and the number of Lyapunov times of reliable prediction is roughly log(1/ε) / log(e). For the Earth's atmosphere, the Lyapunov time is approximately two to three days, setting the fundamental limit on weather prediction. For the solar system, it is tens of millions of years. The Lyapunov time is not a property of the observer or the model; it is an intrinsic time scale of the dynamics itself, independent of units and coordinate choices. In random dynamical systems, the Lyapunov time can be extended or shortened by the coupling between noise and nonlinearity.