17b: What are Lyapunov exponents?
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This article is from the Fractal FAQ, by Ermel Stepp stepp@muvms6.mu.wvnet.edu with numerous contributions by others.
17b: What are Lyapunov exponents?
Lyapunov exponents quantify the amount of linear stability or instability of
an attractor, or an asymptotically long orbit of a dynamical system. 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.
Given two initial conditions for a chaotic system, a and b, which are close
together, the average values obtained in successive iterations for a and b
will differ by an exponentially increasing amount. In other words, the two
sets of numbers drift apart exponentially. If this is written e^(n*(lambda))
for n iterations, then e^(lambda) is the factor by which the distance between
closely related points becomes stretched or contracted in one iteration.
Lambda is the Lyapunov exponent. At least one Lyapunov exponent must
be positive in a chaotic system. A simple derivation is available in:
1. H. G. Schuster, _Deterministic Chaos: An Introduction_, Physics
Verlag, 1984.
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