# 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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