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From: demers@beowulf.ucsd.edu (David E Demers)
Newsgroups: comp.ai.neural-nets
Subject: Re: Lyapunov measures
Message-ID: <7288@sdcsvax.UCSD.Edu>
Date: 23 Oct 89 02:59:46 GMT
References: <173@berlioz.nsc.com> <TED.89Oct21201627@kythera.nmsu.edu>
Sender: nobody@sdcsvax.UCSD.Edu
Reply-To: demers@beowulf.UCSD.EDU (David E Demers)
Organization: EE/CS Dept. U.C. San Diego
Lines: 46

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