Lyapunov exponent

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

:$|\; deltamathbf\{Z\}(t)\; |\; approx\; e^\{lambda\; t\}\; |\; delta\; mathbf\{Z\}\_0\; |$

where $lambda$ is the Lyapunov exponent.

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.

It is named after Aleksandr Lyapunov.

Definition of the maximal Lyapunov exponent

The maximal Lyapunov exponent can be defined as follows:

$lambda\; =\; lim\_\{t\; o\; infty\}\; frac\{1\}\{t\}\; lnfrac\{|delta\; mathbf\{Z\}\_0.$

Definition of the Lyapunov spectrum

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

: $J^t(x\_0)\; =\; left.\; frac\{\; d\; f^t(x)\; \}\{dx\}\; ight|\_\{x\_0\}.$

The $J^t$ matrix describes how a small change at the point $x\_0$ propagates to the final point $f^t(x\_0)$. The limit

: $lim\_\{t\; ightarrow\; infty\}\; (J^t\; cdot\; mathrm\{Transpose\}(J^t)\; )^\{1/2t\}$

defines a matrix $L(x\_0)$ (the conditions for the existence of the limit are given by the Oseledec 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 starting 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.

ignificance 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 knowledgeof the Lyapunov spectrum it is possible to obtain the so-called Kaplan-Yorke dimension $D\_\{KY\}$, 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. $D\_\{KY\}$ represents an upper bound for the information dimension of the system 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)] . Moreover, the sum of all the positive Lyapunov exponents gives an estimate of the Kolmogorov-Sinai entropy accordingly to Pesin's theorem Y. B. Pesin, Characteristic Lyapunov Exponents and Smooth Ergodic Theory, Russian Math. Surveys, 32 (1977), 4, 55-114 ]

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 $L$.

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 G. Benettin, L. Galgani, A. Giorgilli and J.M. Strelcyn, Meccanica, 9-20 (1980); ibidem, Meccanica, 21-30 (1980).] I. Shimada and T. Nagashima, Prog. Theor. Phys. 61, 1605 (1979). ] .

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

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 "x"₀ in phase space. This may be done through the eigenvalues of the Jacobian matrix "J" ⁰("x"₀). These eigenvalues are also called local Lyapunov exponents. The eigenvectors of the Jacobian matrix point in the direction of the stable and unstable manifolds.

ee also

*Aleksandr Lyapunov
*Oseledec theorem
*Liouville's theorem (Hamiltonian)
*Floquet theory
*Recurrence quantification analysis
*Finite Time Lyapunov Exponent
*Finite Size Lyapunov Exponent

References

Further reading

* Cvitanović P., Artuso R., Mainieri R. , Tanner G. and Vattay G. [http://www.chaosbook.org/ Chaos: Classical and Quantum] Niels Bohr Institute, Copenhagen 2005 - "textbook about chaos available under Free Documentation License"

* cite journal
author = Freddy Christiansen and Hans Henrik Rugh
year = 1997
title = Computing Lyapunov spectra with continuous Gram-Schmidt orthonormalization
journal = Nonlinearity
volume = 10
pages = 1063–1072
url = http://www.mpipks-dresden.mpg.de/eprint/freddy/9702017/9702017.ps
doi = 10.1088/0951-7715/10/5/004

* cite journal
author = Salman Habib and Robert D. Ryne
year = 1995
title = Symplectic Calculation of Lyapunov Exponents
journal = Physical Review Letters
volume = 74
pages = 70–73
url = http://arxiv.org/pdf/chao-dyn/9406010
doi = 10.1103/PhysRevLett.74.70

* cite journal
author = Govindan Rangarajan, Salman Habib, and Robert D. Ryne
year = 1998
title = Lyapunov Exponents without Rescaling and Reorthogonalization
journal = Physical Review Letters
volume = 80
pages = 3747–3750
url = http://arxiv.org/pdf/chao-dyn/9803017
doi = 10.1103/PhysRevLett.80.3747

* cite journal
author = X. Zeng, R. Eykholt, and R. A. Pielke
year = 1991
title = Estimating the Lyapunov-exponent spectrum from short time series of low precision
journal = Physical Review Letters
volume = 66
pages = 3229
url = http://link.aps.org/abstract/PRL/v66/p3229
doi = 10.1103/PhysRevLett.66.3229

* cite journal
author = E Aurell, G Boffetta, A Crisanti, G Paladin and A Vulpiani
year = 1997
title = Predictability in the large: an extension of the concept of Lyapunov exponent
volume = 30
pages = 1–26
url = http://www.iop.org/EJ/abstract/0305-4470/30/1/003
journal = J. Phys. A: Math. Gen.
doi = 10.1088/0305-4470/30/1/003

oftware

* [http://www.mpipks-dresden.mpg.de/~tisean/Tisean_3.0.1/index.html] R. Hegger, H. Kantz, and T. Schreiber, Nonlinear Time Series Analysis, TISEAN 3.0.1 (March 2007).

* [http://www.chaoskit.com] Scientio's ChaosKit product calculates Lyapunov exponents amongst other Chaotic measures. Access is provided free online via a web service.

* [ftp://ftp2.sco.com/pub/skunkware/src/x11/misc/mathrec-1.1c.tar.gz] Dr. Ronald Joe Record's mathematical recreations software laboratory includes an X11 graphical client, lyap, for graphically exploring the Lyapunov exponents of a forced logistic map and other maps of the unit interval. The [ftp://ftp2.sco.com/pub/skunkware/src/x11/misc/mathrec-1.1c/ReadMe.html contents and manual pages] of the mathrec software laboratory are also available.

* Jamie Zawinski has incorporated Dr. Record's lyap program into the XScreenSaver collection of free screen savers for X11 and MacOS.