# 17b: What are Lyapunov exponents?

## Description

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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