Lyapunov exponent

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 λ_i are calculated as

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 dL_i(t) in each principal direction. If at least one exponent is positive, this is often an 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.

See also: Lyapunov characteristic number

The following text should be merged in:

The Lyapunov test, also known as the Lyapunov exponent, the methods of approximation by Aleksandr Lyapunov which provide ways of determining the stability of sets of ordinary differential equations (determining the prediction horizon). While there is a whole spectrum of Lyapunov exponents (their number is equal to the dimension of the phase space), the largest is meant; A quantitative measure of the sensitive dependence on the initial conditions; The averaged rate of divergence (or convergence) of two neighboring trajectories; Even qualitative predictions are impossible for a time interval beyond the prediction horizon.

The Lyapunov characteristic exponent [LCE] gives the rate of exponential divergence from perturbed initial conditions.

References (with proper mathematical symbols):