17c: How can Lyapunov exponents be calculated?
Description
This article is from the Fractal FAQ, by Ermel Stepp stepp@muvms6.mu.wvnet.edu with numerous contributions by others.
17c: 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://lyapunov.ucsd.edu/pub/ .
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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