This article is from the sci.fractals FAQ, by Michael C. Taylor and Jean-Pierre Louvet with numerous contributions by others.

For the common periodic forcing pictures, the Lyapunov

exponent is:

lambda = limit as N -> infinity of 1/N times sum from n=1 to N of

log2(abs(dx sub n+1 over dx sub n))

In other words, at each point in the sequence, the derivative of the

iterated equation is evaluated. The Lyapunov exponent is the average

value of the log of the derivative. If the value is negative, the

iteration is stable. Note that summing the logs corresponds to

multiplying the derivatives; if the product of the derivatives has

magnitude < 1, points will get pulled closer together as they go

through the iteration.

MS-DOS and Unix programs for estimating Lyapunov exponents from short

time series are available by ftp: ftp://inls.ucsd.edu/pub/ncsu/

Computing Lyapunov exponents in general is more difficult. Some

references are:

1. H. D. I. Abarbanel, R. Brown and M. B. Kennel, Lyapunov Exponents

in Chaotic Systems: Their importance and their evaluation using

observed data, "International Journal of Modern Physics B" 56, 9

(1991), pp. 1347-1375.

2. A. K. Dewdney, Leaping into Lyapunov Space, "Scientific American",

Sept. 1991, pp. 178-180.

3. M. Frank and T. Stenges, "Journal of Economic Surveys" 2 (1988),

pp. 103- 133.

4. T. S. Parker and L. O. Chua, "Practical Numerical Algorithms for

Chaotic Systems", Springer Verlag, 1989.

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