lyapunov exponents?

Answer 1

The Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories. Quantitatively, two trajectories in phase space with initial separation \delta \mathbf{Z}_0 diverge

| \delta\mathbf{Z}(t) | \approx e^{\lambda t} | \delta \mathbf{Z}_0 |

where | \cdot | represents the modulus of the considered vectors.

The rate of separation can be different for different orientations of initial separation vector. Thus, there is a whole spectrum of Lyapunov exponents—the number of them is equal to the number of dimensions of the phase space. It is common to just refer to the largest one, i.e. to the Maximal Lyapunov exponent (MLE), because it determines the predictability of a dynamical system. A positive MLE is usually taken as an indication that the system is chaotic.

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 estimates the L matrix based on averaging several finite time approximations of the limit defining L.

One of the most used and effective numerical technique to calculate the Lyapunov spectrum for a smooth dynamical system relies on periodic Gram-Schmidt orthonormalization of the Lyapunov vectors to avoid a misalignement of all the vectors along the direction of maximal expansion.

Whereas the (global) Lyapunov exponent gives a measure for the total predictability of a system, it is sometimes interesting to estimate the local predictability around a point x0 in phase space. This may be done through the eigenvalues of the Jacobian matrix J 0(x0). These eigenvalues are also called local Lyapunov exponents. The eigenvectors of the Jacobian matrix point in the direction of the stable and unstable manifolds.
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Answer 2

Lyapunov exponents are used to distinguish between different types of "orbits" in conifguratoin space of dynamical systems. It is a measure of how quickly orbits diverge. Usually denoted by the lamba symbol, if L < 0, its stable, but dissipative and non-conservative. If L = 0, it's stable and conservative. If L > 0, it's unstable and chaotic.