Newsgroups: bit.listserv.frac-l
Date:         Tue, 15 Sep 1992 16:05:33 EDT
From:         McClelland_R%PSB%BLS@NIH3PLUS.BITNET
Subject:      Lyapunov

|Date: 9-15-92  4:00pm
|From: McClelland_R:PSB:BLS
|  To: FRAC-L@GITVM1:BITNET:NIH
|Subj: Lyapunov
|In-Reply-To: Message from FRAC-L@GITVM1:BITNET of 9-15-92
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One way to think about Lyapunov exponents is the following:

Consider two orbits in phase space ("phase space" can be thought of as
a space in which each point represents the condition of the system to be
studied.  For exapmle, suppose you had a mass on a spring.  The phase space
of this system would be a two dimensional plot of position on one axis
and momentum (or velocity, if you like) on the other.  The "orbit" in phase
space is the path that the system follows through that space.  In our
mass-spring example, we might start out with the mass at rest but displaced
from the equilibrium position.  In phase space, this would be represented as
a point on the postion axis (since the momentum is zero).  When we let go of
the mass, the system will gain momentum as the position of the mass
approaches the equilibrium position.  Then, the mass will move past the
equilibrium position and the momentum will begin to decrease again, and
so on.  Through this process, the system will trace out a series of points
in the shape of an ellipse (assuming no friction).  Now is where the fun
begins...)

Suppose we had displaced the mass just a tiny bit further than we did.
What would happen?  Well, we would get another ellipse, just a little
larger than the first one.  Now suppose we watched both of these paths
evolve through time.  Consider the distance between two points that were
initially right next to each other (lets use the starting point, right
where we let go of the mass, as an example).  As time goes on, the points
will remain close together.  This is an example of a nonchaotic system--
the orbits of two points which are initially close will remain close as
time evolves.

A chaotic system, however, would exhibit quite different behavior.  In
a chaotic sytem, the distance between orbits which are initially close
increases exponentially.  This, in fact, is one "definition" of chaos--
i.e., a system which exhibits the above described behavior would be
considered chaotic.

The Lyapunov exponent is a way of quatifying this behavior.  Basically,
we look at the distance between initially close orbits as an exponential
function of time.  If the exponent is zero, we have a nonchaotic system,
as in the mass-spring case.  However, a positive exponent would imply
that the distance is incresing with time, and the system would then be
chaotic.

A nice explanation of this is given in Francis C. Moon's _Chaotic
Vibrations:  An Introduction for Scientists and Engineers_.

Hope this helps.

By the way, I am only a dabbler in this field, so I would welcome
corrections to the above.

Michael South

msouth@grits.valdosta.peachnet.edu
------------------------------------------------------------------------

Michael:

Thanks for the amplifying on my perhaps cryptic comments, especially for
mentioning the embedding of the system in phase space. One qualification
of your answer might be helpful: sensitivity to initial conditions is
not sufficient to describe a chaotic system. The system must also be
deterministic. A stochastic system will exhibit sensitivity as well,
because the forward trajectories will diverge just like a chaotic
system. Chaos might be described as determinism disguising itself as a
stochastic process.

Rob
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