More Fractal Geometry/Chaos Theory

[Matthew P Wiener]

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).  [...]

Eeh...?  This does not have "extreme sensitivity to initial conditions".

>						     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).

Accuracy of 1 meter in this case is *extremely* accurate, since the y
blows up exponentially.  Roughly at the scale of a nucleus on meter
sized objects.  Not a surprise.

>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.

Your comments are complete nonsense.  "The standard restrictions on the
parameters of a dynamical system"?  What the heck are you talking about?

>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)

Gleik's book was actually OK.  Of course, he is not responsible for the
notions of the ignorant who use it as their primary reference.

>that have fixated on the sensitivity to initial conditions as a
>fundamentally nonlinear phenomenon which is just false.

No, it is a fundamentally nonlinear phenomenon.
--
-Matthew P Wiener (weemba at sagi.wistar.upenn.edu)