Lyapunov exponent

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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}|

The rate of separation can be different for different orientations of initial separation vector. Thus, there is 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, because it determines the predictability of a dynamical system.

Definition

For a dynamical system with evolution equation f^{t} in a n–dimensional phase space, the spectrum of Lyapunov exponents

\{\lambda _{1},\lambda _{2},\cdots ,\lambda _{n}\}\,,

in general, depends on the starting point x_{0}. The Lyapunov exponents describe the behavior of vectors in the tangent space of the phase space and are defined from the Jacobian matrix

J^{t}(x_{0})=\left.{\frac  {df^{t}(x)}{dx}}\right|_{{x_{0}}}.

The J^{t} matrix describes how a small change at the point x_{0} propagate to the final point f^{t}(x_{0}). The limit

\lim _{{t\rightarrow \infty }}(J^{t}\cdot {\mathrm  {Transpose}}(J^{t}))^{{1/2t}}

defines a matrix L(x_{0}) (the conditions for the existence of the limit are given by the Oseldec theorem). If \Lambda _{i}(x_{0}) are the eigenvalues of L(x_{0}), then the Lyapunov exponents \lambda _{i} are defined by

\lambda _{i}(x_{0})=\log \Lambda _{i}(x_{0}).\,

The set of Lyapunov exponents will be the same for almost all stating points of an ergodic component of the dynamical system.

Basic properties

If the system is conservative (i.e. there is no dissipation), a volume element of the phase space will stay the same along a trajectory. Thus the sum of all Lyapunov exponents must be zero. If the system is dissipative, the sum of Lyapunov exponents is negative.

If the system is a flow, one exponent is always zero —the Lyapunov exponent corresponding to the eigenvalue of L with an eigenvector in the direction of the flow.

Significance of Lyapunov spectrum

The Lyapunov spectrum can be used to estimate the rate entropy production of the dynamical system.

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, the Lyapunov time will be finite, whereas for regular orbits it will be infinite.

Numerical calculation

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.

See also

References