More Fractal Geometry/Chaos Theory
Ralph Howard
howard at math.SC.EDU
Tue Jul 29 01:21:27 CDT 1997
Juan Cires Martinez writes:
> Having found five minites totally unexpectedly, I'll elaborate a tiny
> bit about the difference between Fractal Geometry and Chaos Theory.
>
> Chaos Theory is the study of dynamical systems that exhibit chaotic
> behavior. Dynamical systems are defined through equations that express
> their evolution in time. If these equations are non-linear, chaotic
> behavior may occur. This chaotic behavior is defined by extreme
> sensitivity to initial conditions, that is, a small change in initial
> conditions of the system (the flapping of wings of the proverbial
> butterfly in Manchuria) induces dramatic changes in behavior (the
> proverbial storm).
<snip of comments on Fractal Geometry>
This brings up one of my pet peeves on how "chaos theory" is presented in
the popular press which is that "extreme sensitivity to initial conditions"
is somehow a non-linear phenomenon. To see this is not the case consider
the easiest linear dynamical system y'(t)=y(t) (sorry for going into nerd
mode and writing equations, but I really want to emphasize how simple the
examples are). This has solution y(t)=y(0)exp(t). If t is time in
seconds and y(t) is measured in meters then and we run the dynamic
system for a minute (t=60) then to get accuracy of 1 meter we need to
measure the initial condition to within 8.75 x 10^(-27)meters which is much
much much smaller than not only an atom (about 10^(-10)meters), but even
the nucleus of an atom (about 10^(-15)meters). Put the other way around,
a error in the initial condition of size of an atom can lead to a error
much larger than the size of the solar system after only a minute for
this LINEAR dynamical system (in fact the error can be as large as about
three times the distance of the sun to the nearest star).
Moreover there is a result that under the standard restrictions on the
parameters of a dynamical system which insure the system will have
solutions existing for all time the one with the greatest sensitivity to
initial conditions is (and you guessed it from the pedantic tone of my
voice) the linear system with the given parameters. Thus in some sense
the linear equations are the MOST sensitive to initial conditions.
Much of the confusion on this subject seems to be due to accounts like
James Gleik's _Chaos_ (a low point in recent popular science books)
that have fixated on the sensitivity to initial conditions as a
fundamentally nonlinear phenomenon which is just false.
Aside:
One place where nonlinearity does make a big difference is in the long time
behavior of a dynamical system. Up until recently (say the last 70 or
so years) the general belief was that if have a solution to a dynamical
system and look at what happens to it in long run (i.e. after a few
thousand years) then either (1) the solution will settle down and
basically stay in one place, (2) the solution will become approximately
periodic, that is rather like the earth around the sun or (3) run off to
infinity in the sense that it just keeps getting farther and farther away.
Linear equations fit into this pattern. However things are not this simple.
For many dynamical systems of interest recently it has been found that
what happens is that in the long run the solution start to move paths on
very complicated objects (the Lorenz mask is an example) that are
fractals in more or less the sense that Juan describes (and Juan I
apologize for choosing your post to rant and rave about, but of the
several on the subject you stated things the most clearly).
To get back to things Pynchonian my guess as to why he does not use
chaos theory in M&D is more based on literary rather than scientific
concerns. At least in GR when mathematical ideas are used as metaphors
(rather than just word play) they are used very precisely and correctly.
For example his analysis of flight (GR page 567), from canon balls to
rockets ("Three hundred years ago mathematicians were learning to break the
canonball's rise and fall into stair steps of range and height Deltax and
Deltay, allowing them to approaching zero as armies of eternally shrinking
midgets galloped upstairs and down again, the patter of there little feet
growing finer, smoothing out into continuous sound"), is written by
someone who knows the foundations of the calculus very well indeed (this
sentence goes very much past name dropping). Moreover his mathematical
metaphor is a help for me, and I assume others that are familiar with
mathematics, in understanding his points about history and "... film and
calculus, both pornographies of flight". From past posts to the list I
have the impression that other of Pynchon's scientific metaphors, at
least in GR, are equally precise and to the point.
However references to "Chaos Theory" do not call up precise images and
are therefore pretty much useless as metaphors in throwing light on other
subjects.
Ralph
--
Ralph Howard Phone: (803) 777-2913
Department of Mathematics Fax: (803) 777-3783
University of South Carolina e-mail: howard at math.sc.edu
Columbia, SC 29208 USA http://www.math.sc.edu/~howard/
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