Lyapunov and chaos theory
Date: Around 1995
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
Try the book: Chaotic and Fractal Dynamics by F. C. Moon, published by John
Wiley & Sons, 1992.
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).
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Update: June 2012