Lyapunov exponents
Chaotic systems are unpredictable. Or rather chaotic systems are not deterministically predictable in the long run.
You can make predictions if you weaken one of these requirements. You can make deterministic predictions in the short run, or statistical predictions in the long run.
Lyapunov exponents are a way to measure how quickly the short run turns into the long run. There is a rigorous definition of a Lyapunov exponent, but I often hear the term used metaphorically. If something is said to have a large Lyapunov exponent, it quickly becomes unpredictable. (Unpredictable as far as deterministic prediction.)
In a chaotic system, the uncertainty around the future state of the system grows exponentially. If you start with a small sphere of initial conditions at time t = 0, a sphere of diameter , that sphere becomes an ellipsoid over time. The logarithm of the ratio of the maximum ellipsoid diameter to the diameter of the initial sphere is the Lyapunov exponent . (Technically you have to average this log ratio over all possible starting points.)
With a Lyapunov exponent the uncertainty in your solution is bounded by exp(t).
You can predict the future state of a chaotic system if one of , , or t is small enough. The larger is, the smaller t must be, and vice versa. So in this sense the Lyapunov exponent tells you how far out you can make predictions; eventually your error bars are so wide as to be worthless.
A Lyapunov exponent isn't something you can easily calculate exactly except for toy problems, but it is something you can easily estimate empirically. Pick a cloud of initial conditions and see what happens to that cloud over time.
Related posts- Does chaos imply period 3?
- Everywhere chaotic map on the sphere
- 3D bifurcation diagram
- The other butterfly effect