

Lyapunov and chaos theory
Name: jcolombe
Status: N/A
Age: N/A
Location: N/A
Country: N/A
Date: Around 1995
Question:
I am trying to understand a little more about chaos theory than Gleick wrote
about. Can someone explain Lyapunov analysis of stability? Lyapunov
exponents? If I have the output of a system through time, can I guess at
how convergent its behavior is (i.e. how "unchaotic", and thus how chaotic)?
I am trying to hack through Nemytski's "Qualitative Theory of Differential
Equations" but it reads like Greek.
By the way, I am a computational neurobiologist in search of what NOVELTY
looks like to a dynamic system like the brain. For that matter, what does
FAMILIARITY look like? I think it has got something provocative to do with
chaos!
Replies:
Try the book: Chaotic and Fractal Dynamics by F. C. Moon, published by John
Wiley & Sons, 1992.
hawley
Computational neurobiology. Wow! Sounds exciting! To actually answer the
question though, suppose you have a chaotic (or otherwise) system that is
described by some set of coordinates we can label by the single symbol x.
The system evolves in time in some way  the most common fashion is with
discrete time steps, so x > x' > x" etc., but continuous time works
just as well (changes the meaning of exponents though). In the discrete
time case, suppose we start with two points x1 and x2, a "distance" d
apart (normally d = square root of sum of squares of differences in x1 and
x2, but could be any meaningful measure of distance). Then evolving forward
in time, x1 > x1' > x1" . . . and x2 > x2' > x2" . . . after n
steps, measure the difference between x2(n) and x1(n) and produce a distance
d(n). If the distance d(n) grows geometrically like d(0) e^lambda n, then
the system has a positive Lyapunov exponent lambda. If d(n) shrinks with
time like d(0) e^lambda n, then the Lyapunov exponent lambda is negative.
The bigger lambda is the more chaotic (in the positive case) or unchaotic
(in the negative case).
asmith
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Update: June 2012

