The Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a measure that determines for each point of phase space, how quickly trajectories that begin in this point diverge over time.
Actually, the number of Lyapunov exponents is equal to the number of dimensions of the embedding phase space, but it is common to just refer to the largest one, because it determines the predictability of a dynamical system.
The Lyapunov exponents λi are calculated as
- <math>\lambda_i = \lim_{t \to \infty} \frac{1}{t} \ln \left( \frac{d L_i(t)}{d r} \right)</math>,
which can be thought of as following the motion of an infinitesimally small sphere, with an initial radius dr, that starts from the point for which the exponent should be calculated. On its trajectory, it will get "squished" unevenly, so that it becomes an ellipsoid with time-dependent radiuses dLi(t) in each principal direction.
The Lyapunov exponents are nothing else than the long-term exponential growth rates for systems starting from the given point. If at least one exponent is positive, this is often in indication that the system is chaotic.
Note however, that phase space volume does not necessarily change over time. If the system is conservative (i.e. there is no dissipation), volume will stay the same.
