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39 How can Lyapunov exponents be calculated?




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

39 How can Lyapunov exponents be calculated?

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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