I see the words 'chaos' and 'fractals' very often here.
and somehow I get a vague impression, that they are some kind of buzz words, that
have become popular simpley because they have a certain 'sex appeal' a
look good on a graphics screen. Some (several) years ago, when I last
read anything about these things, was in 'The mathematical Intelligenser'
from Springer, and what I read was a not very kind article about
Mandelbrot and the fuzz about fractals. So what is the truth about it?
Chaos theory: a system behaves chaotically if it behaves reproducibly
given the same initial conditions, given slightly different initial
conditions, it behaves very differently. In other words, it is very
sensitive to initial conditions. This is completely distinct from
randomness or noise.
fractals: popularized by Mandelbrot who treated them like a business
instead of science for which he was criticized. We have all seen the
pretty pictures. The essential ingredient is self-similarity which
means the pattern looks the same no matter what size scale you use.
fuzzy logic: (I am a truck) if I am going 20 MPH and the driver is applying
the brakes, then I should apply light braking pressure with a probability
of 20%, medium pressure with a probability of 50%, and high pressure
with a probability of 30%.
The important thing about chaos (and to a lesser extent
fractals) is that they give us a new mathematically based picture
of certain ways in which the real world behaves, which is probably
obvious to most people without the mathematics. The old ideas
(since Newton himself!) were that the universe ran like clockwork -
if you knew well enough what everything in the universe was
doing at a particular time, you could predict the future
of everything arbitrarily far. That is still sort of true (barring
randomness from quantum mechanics) - but chaos theory tells us
the immense precision that would be required to do that sort
of precision - basically it is the reason we will probably never
be able to predict the weather more than a week in advance, unless
we control it in some way... Chaos theory (and fractlas) tell
us more though - that even with this hopelessness of prediction
of specifics (where every single particle will be indefinitely into
the future) we still can get pictures of patterns that tell us
qualitatively what will happen, and that is better than nothing!
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Update: June 2012