### Chaotic Dynamics and Fractals

This is a "hot" topic, or at least it has been in the last ten years. For some
very nice fractal pictures, this site in Vancouver (so the loading time may be
a while) is unbeatable:
Daryl Hepting's Art Gallery
The related topic of * chaos * is studied in our department; we consider
the reliability of scientific computing methods for chaotic dynamical systems, and
applications of chaotic dynamics to assess the reliability of scientific computing
methods to predict (or falsely predict) global warming. We look at ways of predicting
when chaotic behaviour can occur (such as in oscillation of structures) so it can
be avoided, and of applying chaotic behaviour to chemical and molecular systems
when it produces useful results (such as in mixing of materials).
But what is "chaos"? In the previous century, * linear * mathematical
models were used to predict the dynamical behaviour of objects; these linear models
had solutions that one could write down explicitly, like exp(-0.1*t)*sin(t).
You tell me the time
t, and I can tell you what exp(-0.1*t)*sin(t) is---the answer is perfectly
predictable, and if
you only give me three places of accuracy in t then I can give you three places of
accuracy in the answer (in the absolute error sense).

But for chaotic systems, the situation is very different. There are *no*
formulas that anyone could write down for the solution (usually); instead we have
computer programs such as

x = input
for i=1:100,
x = 3.9*x*(1-x)
end;
output x

for which incredibly tiny changes in the input produce drastic, chaotic, changes in
the output. Though the system is perfectly deterministic, not random at all, the
fact that our knowledge of the input is imperfect destroys the predictability of the
outcome.