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2.11] What are Lyapunov exponents? (nonlinear science)


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

2.11] What are Lyapunov exponents? (nonlinear science)

(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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