More Fractal Geometry/Chaos theory

[Sanjay Krishnaswamy]

Ummm, sorry for the theoretical digression but. . . . 
Ralph Howard writes:

>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).

This misses the point entirely.  For the linear system -- by definition! --
_arbitrarily close_ initial conditions will produce _arbitrarily close_
results.  (Or, to be formal, invoke the "for eny epsilon greater than zero
there exists a delta such that f(x+delta) - f(x) < epsilon" shtick -- proof
left to reader!) Is an error the size of the solar system too large?  Then
increase the precision and accuracy of your original measurement.  "Extreme
sensitivity to initial conditions" _does_ mean something -- it means no
matter how small your error, it means that the results at some later time t
are _completely unpredictable_.  You can't even, as you can in your
example, say, "well, my error in the inital measurement is this big, so the
spread in what I expect at time t is this."  The inital error leave you
lost.  Whatever you may think of Gleick, this _is_ a fundamental and
_unique_ property of nonlinear systems, in that a _linear_ system is
_necessarily_ one in which, as you match the inital conditions more
closely, you must match the long-term behavior more closely as well (there
are more formal and general formulations of this idea and I'd be happy to
discuss them off the list).  It is also a crucial property of those systems
from a dynamical systems perspective, because it has helped us determine
which aspects of a system are worth examining -- for example in the (pretty
much solved) problem of a turbulent jet, where the scientist goes from the
strain tensor and formulates generalities about the momentum and pressure
field without attempting to define steady-flow properties.  

Again, sorry for the non-literary digression.

____________________________________________________________________________
Sanjay Krishnaswamy
sanjay at visidyne.com