More Fractal Geometry/Chaos theory

[Sanjay Krishnaswamy]

In response to Mr. Dinn's question, briefly, in case anyone else is
interested:

Sorry, I should have avoided the attempt to be cute.  Linear systems can be
discontinuous (the classic example, which you might find in some textbooks,
is infinitely pointy everywhere -- you can even _make_ functions which are
infinitely pointy, though linear combinations of nice, smooth functions.
Nifty?  I sure as heck think so.)  or continuous.  Non-linear systems can
be discontinuous, or continuous.  Epsilons and deltas reoccur in a lot of
formal mathematical proofs that one remembers vaguely from highschool
because, well, what the hell, there are only so many Grrek letters!  So,
"linear" and "continuous" -- no relation.  What "linear" usually means
(although in different contexts it has analogous but different senses) is
that if you plug in one thing and get one answer, then you plug in
something else and get another answer, well then, what you get if you plug
in the sum of the original two arguments is the sum of the original
answers.  Which probably doesn't jibe at first with your intuitive sense of
"linear" -- it sort of means that, if a function "stretches out" some
space, then it does it nice and evenly.  But I'm sure there are people who
can phrase that better.  Anyway: where that impacts what I said earlier is,
as you increase the argument to a linear function by things close to zero,
you have little effect upon the function's output -- how close is "close"
and how little is "little" is where the epsilon-delta jazz comes in since
Mr. Howard's original pointing justly pointed out that for some functions,
"close" had better be pretty damn close, for "little" to be little!  But
still -- if the function is _linear_, you can always get close enough.  If
it's _not_, you can't.  I think that is understandable in lay English as
"extreme sensitivity to initial conditions"  -- the point is, the error in
your inital conditions makes prediction of _certain_ system properties moot
-- not even something you can predict "with error bars" so to speak.
Whether or not the phase "extreme sensitivity to initial conditions" is the
best phrase to describe that phenomenon, is a question of semantics, and
I'm not qualified to address it.  Also note that not all nonlinear
functions behave like that -- the voltage out of a CMOS chip, for example,
might look sort of like
	f(x) = 0 if x < 2.5
	f(x) = 5 if x > 2.5
which is pretty nonlinear, and nonetheless -- well, if you're pretty sure
you're putting in 5 volts, you're gonna get 5 volts out.  So, forgive me if
I've made some sweeping generalizations in my previous post in my haste to
correct a point.

SK
____________________________________________________________________________
Sanjay Krishnaswamy
sanjay at visidyne.com