andrew@dtg.nsc.com (Lord Snooty @ The Giant Poisoned Electric Head ) (10/10/89)
Can anyone suggest texts which discuss Lyapunov functions/measures? I know very little about them, except that they seem to be popular in describing global stability criteria (e.g. Cohen & Grossberg, Hopfield) and so would be a useful addition to the NN analyst's toolkit. Unfortunately, I've come across no books which deal with this on a level accessible to undergrad maths. Thanks in advance, -- ........................................................................... Andrew Palfreyman and the 'q' is silent, andrew@dtg.nsc.com as in banana time sucks
worden@ut-emx.UUCP (worden) (10/21/89)
Lyapunov functions and stability criteria are one of the mainstays of control theory (aka, linear systems theory). There are many, many control theory texts ... at both the undergraduate and graduate levels ... and throughout the spectrum from pure applied to pure theorem-proof. An hour or two spent browsing in your local engineering library would be an excellent starting place. - Sue Worden Electrical & Computer Engineering University of Texas at Austin
ted@nmsu.edu (Ted Dunning) (10/22/89)
In article <173@berlioz.nsc.com> andrew@dtg.nsc.com (Lord Snooty @ The Giant Poisoned Electric Head ) writes:
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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
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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 Mondenscheindemers@beowulf.ucsd.edu (David E Demers) (10/23/89)
In article <TED.89Oct21201627@kythera.nmsu.edu> ted@nmsu.edu (Ted Dunning) writes: > >In article <173@berlioz.nsc.com> andrew@dtg.nsc.com (Lord Snooty @ The Giant Poisoned Electric Head ) writes: >> 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. [discussion of lyapunov exponents deleted] >ted@nmsu.edu Actually, I suspect the poster DID mean Lyapunov functions, NOT Lyapunov exponent... both stem from Lyapunov's :-) work around the turn of the century. A Lyapunov function is some function which obeys a number of properties. I have not seen any general introduction to Lyapunov functions, but haven't looked hard either. The key properties of a Lyapunov function for a physical system are that: 1. the function is bounded from below (or with a change of sign, from above...), 2. (some smoothness criteria...) 3. It can be shown that any state change the system undergoes results in the value of the Lyapunov function not increasing. Thus the system will eventually reach a stable state (could be a limit cycle, there may be more than one point at which the function reaches a minimum). In the context of neural nets, Hopfield showed that his "Energy" function is a Lyapunov function for a Hopfield net, thus proving that such a net will eventually reach an equilibrium. Kosko proved essentially the same result for the BAM. The trick, of course, for any system is FINDING a Lyapunov function. If you can show isomorphism with some physical system for your abstract system, then perhaps you can use known properties of the physical system by analogy. Dave
enorris@gmuvax2.gmu.edu (Gene Norris) (10/24/89)
In article <7288@sdcsvax.UCSD.Edu> demers@beowulf.UCSD.EDU (David E Demers) writes: (stuff deleted) >>> Can anyone suggest texts which discuss Lyapunov functions/measures? A classic textbook is LaSalle and Lefshetz, Stability Theory of Ordinary Differential Equations. A more current reference is Morris Hirsh's lovely paper in Neural Networks, vol2 no. 5, (1989). Eugene Norris CS Dept George Mason University Fairfax, VA 22032 (703)323-2713 enorris@gmuvax2.gmu.edu