Lyapunov exponent

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In mathematics 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.

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[edit] Definition of the maximal Lyapunov exponent

The maximal Lyapunov exponent can be defined as follows:

\lambda = \lim_{t \to \infty} \lim_{|\delta \mathbf{Z}_0| \to 0} \frac{1}{t} \log\frac{| \delta\mathbf{Z}(t)|}{|\delta \mathbf{Z}_0|}

the order of the limits should be preserved to have a meaningful definition. Therefore the MLE is defined as the exponential rate of separation of a reference orbit with respect to an infinitesimally perturbed orbit averaged over a an extremely long (infinite) lag of time.

[edit] Definition of the Lyapunov spectrum

For a dynamical system with evolution equation ft 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 x0. 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{ d f^t(x) }{dx} \right|_{x_0}.

The Jt matrix describes how a small change at the point x0 propagates to the final point ft(x0). The limit

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

defines a matrix L(x0) (the conditions for the existence of the limit are given by the Oseldec theorem). If Λi(x0) are the eigenvalues of L(x0), then the Lyapunov exponents λi are defined by

\lambda_i(x_0) = \log \Lambda_i(x_0).\,

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

[edit] 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.

[edit] Significance of the Lyapunov spectrum

The Lyapunov spectrum can be used to give an estimate of the rate of entropy production and of the fractal dimension of the considered dynamical system. In particular from the knowledge of the Lyapunov spectrum it is possible to obtain the so-called Kaplan-Yorke dimension DKY, that is defined as follows:

D_{KY}= k + \sum_{i=1}^k \lambda_i/|\lambda_{k+1}|,

where k is the maximum integer such that the sum of the k largest exponents is still non-negative. DKY represents an upper bound for the information dimension of the system [1]. Moreover, the sum of all the positive Lyapunov exponents gives an estimate of the Kolmogorov-sinai entropy accordingly to the Pesin's theorem.

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.

[edit] 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.

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 [2].

For the calculation of Lyapunov exponents from limited experimental data, various methods have been proposed.

[edit] Local Lyapunov exponent

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.

[edit] See also

[edit] References

  1. ^ J. Kaplan and J. Yorke Chaotic behavior of multidimensional difference equations In Peitgen, H. O. & Walther, H. O., editors, ``Functional Differential Equations and Approximation of Fixed Points Springer, New York (1987)
  2. ^ G. Benettin, L. Galgani, A. Giorgilli and J.M. Strelcyn;Meccanica, 9-20 (1980); ibidem, Meccanica, 21-30 (1980).


[edit] Software

  • [1] R. Hegger, H. Kantz, and T. Schreiber, Nonlinear Time Series Analysis, TISEAN 2.1 (December 2000).
  • [2] Scientio's ChaosKit product calculates Lyapunov exponents amongst other Chaotic measures. Access is provided free online via a web service.
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