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

$\lambda_i = \lim_{t \to \infty} \frac{1}{t} \ln \left( \frac{d L_i(t)}{d r} \right),$

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 radii 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.

Another commonly used definition for Lyapunov exponents is given by

$\lambda_i = \lim_{t \to \infty} \lim_{\left\| \delta Z_0 \right\| \to 0} \frac{1}{t} \ln \left( \frac{\left\| \delta\mathbf{Z}(t) \right\|}{\left\| \delta \mathbf{Z}_0 \right\|} \right),$

where $\delta \mathbf{Z}_0$ is the initial distance in phase-space between a reference solution and a perturbation, and $\delta \mathbf{Z}(t)$ is the distance at time t. For each choice of direction for the initial perturbation a different Lyapunov exponent is given.

The inverse of the largest Lyapunov exponent is sometimes referred in literature as Lyapunov time, and defines the characteristic e-folding time. For chaotic orbits, Lyapunov time will be finite, whereas for regular orbits it will be infinite.

Generally the calculation of Lyapunov exponents, as defined above, cannot be carried out analytically, and in most cases one must resort to numerical techniques. The commonly used numerical procedures calculate the largest Lyapunov exponents and recover the others by means of the Gram-Schmidt process.

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.

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Categories: Dynamical systems