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.