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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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