| MadSci Network: Other |
Chaos Theory is a branch of mathematics which, depending on the way you're
going, either delves deeper into abstract mathematics or applies better to
nature than any other branch. The back cover of James Gleick's book
Chaos: Making of a New Science describes Chaos Theory as "a way of
seeing order where formerly only the random, the erratic, the
unpredictable--in short, the chaotic--had been observed." The main focus
of the branch is dynamical systems. That is, situations with sensitive
dependence on initial conditions. What this proves about mathematics is
that the more abstract mathematics gets, the better it can be applied to
the real world. Chaos Theory is often affectionately called the "Science
of Complexity," or the "Science of Surprise."
For example, the spread of disease can be modelled by a relatively simple
function called the
Logistics Equation. The spread of disease depends on
many things: The number of people initially infected, the number of
carriers (people who have the virus but do not show symptoms), the number
of people not infected, whether or not the disease is curable, etc..
Depending on where you start you might end up in a place where there is a
fixed number of people infected (such as when no one is infected in the
first place) or with two fixed numbers of people infected, or with wildly
differing numbers of people infected from minute to minute.
My personal favorite example of a chaotic system is this: Take a small,
fairly heavy object, such as a lead sinker or heavy ring and tie a long,
skinny elastic to it and hold the assembly by the other end of the
elastic. First, just pull it gently to one side and release. This should
give you a nice, even pendular motion which is easily modelled by a simple
equation. Now, pull to one side, stretching the elastic, and bobbing the
your arm up and down. Look at it go wild! Mathematicians are still
working on modelling that movement!
An example of the theory can probably be found in your favorite place to
buy posters. It's called a fractal and the patterns of the very detailed
pictures are really pretty. Fractals illustrate sensitivity to initial
conditions on the complex plane. (I don't know if you've had complex
numbers yet, but you can ask me about those seperately.) What happens to
make a fractal is that a function (such as x^2 - x or whatever) is applied
to each point on the plane repeatedly. Some points will stay as they are,
such as zero, which will go: 0, 0, 0... Others will "orbit" and go
between two values. Others will blast off into infinity, but usually at
different rates. Usually to color fractals a maximum value is chosen and
points are colors according to how quickly a point exceeds the chosen
value.
I highly recommend visiting the following websites which have fun toys to
play with:
Try the links in the MadSci Library for more information on Other.