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

|\delta \mathbf {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. Note that an arbitrary initial separation vector will typically contain some component in the direction associated with the MLE, and because of the exponential growth rate the effect of the other exponents will be obliterated over time.

It is named after Aleksandr Lyapunov.

Definition of the maximal Lyapunov exponent

The maximal Lyapunov exponent can be defined as follows:

$\lambda =\lim _{t\to \infty }{\frac {1}{t}}\ln {\frac {|\delta \mathbf {Z} (t)|}{|\delta \mathbf {Z} _{0}|}}.$

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 $x_{0}$ . (However, we will usually be interested in the attractor (or attractors) of a dynamical system, and there will normally be one set of exponents associated with each attractor. The choice of starting point may determine on which attractor the system ends up on, if there is more than one.) 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 $x_{0}$ propagates 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 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 and the trajectory does not converge to a single point, 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 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 $D_{KY}$ , 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.^[1] Moreover, the sum of all the positive Lyapunov exponents gives an estimate of the Kolmogorov–Sinai entropy accordingly to Pesin's theorem^[2]

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. An early example, which also constituted the first demonstration of the exponential divergence of chaotic trajectories, was carried out by R. H. Miller in 1964.^[3] Currently, the most commonly used numerical procedure 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 misalignment of all the vectors along the direction of maximal expansion ^[4] ^[5].

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 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. (A word of caution - unlike the global exponents, these local exponents are not invariant under a nonlinear change of coordinates.)

Conditional Lyapunov exponent

This term is normally used in regards to the synchronization of chaos, in which there are two systems that are coupled, usually in a unidirectional manner so that there is a drive (or master) system and a response (or slave) system. The conditional exponents are those of the response system with the drive system treated as simply the source of a (chaotic) drive signal. Synchronization occurs when all of the conditional exponents are negative. See, e.g., see Pecora et al, Chaos Vol. 7, No. 4, (1997), 520.

References

^ 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)
^ Y. B. Pesin, Characteristic Lyapunov Exponents and Smooth Ergodic Theory, Russian Math. Surveys, 32 (1977), 4, 55-114
^ R. H. Miller, Irreversibility in Small Stellar Dynamical Systems, The Astrophysical Journal, 140, 250 (1964)
^ 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).

Software

[1] R. Hegger, H. Kantz, and T. Schreiber, Nonlinear Time Series Analysis, TISEAN 3.0.1 (March 2007).

[2] Scientio's ChaosKit product calculates Lyapunov exponents amongst other Chaotic measures. Access is provided online via a web service and Silverlight demo .

[3] 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 contents and manual pages of the mathrec software laboratory are also available.

[4] On this web page is software called LyapOde, which includes source code written in "C" for calculation of Lyapunov exponents when the differential equations are known. It can also calculate the conditional Lyapunov exponents for coupled identical systems. The software can be compiled for running on Windows, Mac, or Linux/Unix systems. It includes the equations for several systems (Lorenz, Rossler, etc.) and documentation on how to create and compile software for additional systems of the user's choice. The software runs in a text window and has no graphics capabilities, but it is efficient and has no inherent limitations on the number of variables etc. It uses the QR decomposition method of Eckmann and Ruelle (Reviews of Modern Physics, Vol. 57, No. 3, Part1, (1985), 617).

[ky-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)

[pesin-2] Y. B. Pesin, Characteristic Lyapunov Exponents and Smooth Ergodic Theory, Russian Math. Surveys, 32 (1977), 4, 55-114

[3] R. H. Miller, Irreversibility in Small Stellar Dynamical Systems, The Astrophysical Journal, 140, 250 (1964)

[benettin-4] G. Benettin, L. Galgani, A. Giorgilli and J.M. Strelcyn, Meccanica, 9-20 (1980); ibidem, Meccanica, 21-30 (1980).

[shimada-5] I. Shimada and T. Nagashima, Prog. Theor. Phys. 61, 1605 (1979).

[1]

[2]

[3]

[4]

[5]