This article is from the Nonlinear Science FAQ, by James D. Meiss jdm@boulder.colorado.edu with numerous contributions by others.

(Thanks to Ronnie Mainieri & Fred Klingener for contributing to this answer)

The hardest thing to get right about Lyapunov exponents is the spelling of

Lyapunov, which you will variously find as Liapunov, Lyapunof and even

Liapunoff. Of course Lyapunov is really spelled in the Cyrillic alphabet:

(Lambda)(backwards r)(pi)(Y)(H)(0)(B). Now that there is an ANSI standard of

transliteration for Cyrillic, we expect all references to converge on the

version Lyapunov.

Lyapunov was born in Russia in 6 June 1857. He was greatly influenced by

Chebyshev and was a student with Markov. He was also a passionate man:

Lyapunov shot himself the day his wife died. He died 3 Nov. 1918, three days

later. According to the request on a note he left, Lyapunov was buried with

his wife. [biographical data from a biography by A. T. Grigorian].

Lyapunov left us with more than just a simple note. He left a collection of

papers on the equilibrium shape of rotating liquids, on probability, and on

the stability of low-dimensional dynamical systems. It was from his

dissertation that the notion of Lyapunov exponent emerged. Lyapunov was

interested in showing how to discover if a solution to a dynamical system is

stable or not for all times. The usual method of studying stability, i.e.

linear stability, was not good enough, because if you waited long enough the

small errors due to linearization would pile up and make the approximation

invalid. Lyapunov developed concepts (now called Lyapunov Stability) to

overcome these difficulties.

Lyapunov exponents measure the rate at which nearby orbits converge or

diverge. There are as many Lyapunov exponents as there are dimensions in the

state space of the system, but the largest is usually the most important.

Roughly speaking the (maximal) Lyapunov exponent is the time constant, lambda,

in the expression for the distance between two nearby orbits, exp(lambda *

t). If lambda is negative, then the orbits converge in time, and the

dynamical system is insensitive to initial conditions. However, if lambda is

positive, then the distance between nearby orbits grows exponentially in time,

and the system exhibits sensitive dependence on initial conditions.

There are basically two ways to compute Lyapunov exponents. In one way one

chooses two nearby points, evolves them in time, measuring the growth rate of

the distance between them. This is useful when one has a time series, but has

the disadvantage that the growth rate is really not a local effect as the

points separate. A better way is to measure the growth rate of tangent vectors

to a given orbit.

More precisely, consider a map f in an m dimensional phase space, and its

derivative matrix Df(x). Let v be a tangent vector at the point x. Then we

define a function

1 n L(x,v) = lim --- ln |( Df (x)v )| n -> oo n

Now the Multiplicative Ergodic Theorem of Oseledec states that this limit

exists for almost all points x and all tangent vectors v. There are at most m

distinct values of L as we let v range over the tangent space. These are the

Lyapunov exponents at x.

For more information on computing the exponents see

Wolf, A., J. B. Swift, et al. (1985). "Determining Lyapunov Exponents from a

Time Series." Physica D 16: 285-317.

Eckmann, J.-P., S. O. Kamphorst, et al. (1986). "Liapunov exponents from

time series." Phys. Rev. A 34: 4971-4979.

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