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38 What are Lyapunov exponents?




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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.

38 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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