Chaos

Date: 01/27/2001 at 14:27:39
From: Justin
Subject: Chaos

Edward Lorenz came up with the branch called "Chaos", and The 
Butterfly effect. How can a butterfly flapping its wings in South 
America cause a tornado in Texas?

Date: 01/27/2001 at 16:42:38
From: Doctor Douglas
Subject: Re: Chaos

Hi Justin, and thanks for sending your question to Ask Dr. Math.

Here'a different example, like the butterfly scenario, that 
illustrates the "sensitive dependence on initial conditions" that is 
the trademark of chaos.

I'll assume that you're familiar with (American) football. If you're
not, write back and I'll try to come up with a different example.
Suppose a football is punted and is oriented so that the points are up
and down. It is falling, and headed right for the 50 yd line. The 
football is not spinning. The punt returner decides that it is too 
dangerous to field, and lets the ball bounce. Which way will it 
bounce?  

The answer is that it depends. If the downwards point of the football 
is just a little bit to one side, it will tend to bounce away from 
that side, say, towards the goal of the home team. But if the point is 
just a little bit to the other side, it will bounce toward the other 
goal. And if the point is just a little to the side, it will bounce 
towards one of the two benches. In this way a tiny difference in the 
position, or the movement of the air molecules around the ball that 
influence its orientation on the way down could have a big difference 
on where the ball ends up. The direction of a small 1 mph breeze might 
make a difference of 30 yards in where the football ends up.

It is in this spirit that the butterfly's flapping its wings might 
affect things many days later, many miles away, and either create a 
tornado or suppress one from forming.

I hope this helps clarify the concept of "sensitive dependence on 
initial conditions." If you have more questions about the fascinating 
topic of mathematical chaos, please write back.

- Doctor Douglas, The Math Forum
  http://mathforum.org/dr.math/