Path: utzoo!utgpu!jarvis.csri.toronto.edu!mailrus!cs.utexas.edu!rutgers!cmcl2!lanl!opus!ted From: ted@nmsu.edu (Ted Dunning) Newsgroups: comp.ai.neural-nets Subject: Re: Lyapunov measures Message-ID: Date: 22 Oct 89 02:16:27 GMT References: <173@berlioz.nsc.com> Sender: news@nmsu.edu Organization: NMSU Computer Science Lines: 50 In-reply-to: andrew@dtg.nsc.com's message of 10 Oct 89 00:14:27 GMT In article <173@berlioz.nsc.com> andrew@dtg.nsc.com (Lord Snooty @ The Giant Poisoned Electric Head ) writes: Path: opus!lanl!cmcl2!nrl-cmf!ames!uakari.primate.wisc.edu!brutus.cs.uiuc.edu!apple!voder!dtg.nsc.com!andrew From: andrew@dtg.nsc.com (Lord Snooty @ The Giant Poisoned Electric Head ) Newsgroups: comp.ai.neural-nets Keywords: biblios Date: 10 Oct 89 00:14:27 GMT Organization: National Semiconductor, Santa Clara Lines: 12 Can anyone suggest texts which discuss Lyapunov functions/measures? look in texts on chaos. guckenheimer and holmes is the most complete, but it is hardly accessible or obvious. devaney's book on chaos must mention lyapunov exponents and it is certainly accessible. Unfortunately, I've come across no books which deal with this on a level accessible to undergrad maths. and you may not depending on what you think of as undergraduate math. but here is a quick summary anyway... the idea behind a lyapunov exponent (not measure), is that any smooth flow defined by differential equations or an iterated map will transform some small neighborhood around a fixed point to some other small neighborhood around that same point. if the original neighborhood is small enough, then the transformation will be approximately linear (remember, a smooth flow or continous and differentiable map). this transformation can be described by the directions in which stretching occurs and the amount of stretching. the directions are called the eigenvectors and the amounts are called the eigenvalues of the transformation. the natural log of the largest of these eigenvalues is called the lyapunov exponent. if it is negative then the fixed point is stable since the neighborhood around it must contract toward the fixed point. if it is positive, then the fixed point is at best a hyperbolic fixed point with some flow toward it and some away (i.e. it is at a saddle bifurcation point in the flow). in the case of rotating flow, then we use what are called generalized eigenvalues. they still capture the essence of growth or shrinkage, but not along a single axis. -- ted@nmsu.edu Dem Dichter war so wohl daheime In Schildas teurem Eichenhain! Dort wob ich meine zarten Reime Aus Veilchenduft und Mondenschein