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. hawleyComputational 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 Click here to return to the Mathematics Archives

NEWTON is an electronic community for Science, Math, and Computer Science K-12 Educators, sponsored and operated by Argonne National Laboratory's Educational Programs, Andrew Skipor, Ph.D., Head of Educational Programs.

For assistance with NEWTON contact a System Operator (help@newton.dep.anl.gov), or at Argonne's Educational Programs