Chaotic Functions

Date: 10/30/2000 at 21:06:09
From: Suzie
Subject: Chaos Theory

How can you explain the chaos theory mathematically?

Date: 10/31/2000 at 02:15:10
From: Doctor Schwa
Subject: Re: Chaos Theory

There are lots of parts of the theory, many of which require some 
calculus to understand. There's a very nice book that explains a lot 
of it pictorially, but you still need to know some calculus: 
_Dynamics: The Geometry of Behavior_ by Ralph H. Abraham and 
Christopher D. Shaw:

  http://www.dakota-books.com/Blurbs/dyn.html   

One of the key parts of chaos theory is pretty easy to understand, 
though. It's often called the "butterfly effect": a butterfly flapping 
its wings in China can have a big effect on the San Francisco weather 
a few months later. That is, an infinitesimally tiny change at one 
point in time can render things completely unpredictable down the 
road.

One way to model this mathematically is with the function

     f(x) = fractional part of 10x

That is, start with any number, for instance:

     0.142857129819234609814

Multiply by 10, and drop the whole-number part:

     0.42857129819234609814

and again, getting:

     0.2857129819234609814

and so on. You can see that if there's a tiny, tiny change in the 
original number, to:

     0.142857129819234609815

instead, after a certain number of steps we'll have the HUGE change of 
0.4 vs. 0.5 ...what was a 0.000000000000000001% change or so is now a 
20% difference after only 20 steps.

For another example, try the function that outputs your number, times 
(1 - the number), times a constant k. Mathematically:

     f(x) = x*(1-x)*k

If your constant k is 2.1, for instance, and your starting number is 
0.5, then after a few steps you get 0.523809524. In fact no matter 
what your starting number is, you end up at that number pretty fast, 
if k is 2.1. This is called an "attractor." It's like falling down a 
hill.

Now try the same thing with k = 2.5, or 2.8 ...you'll approach 
different numbers (0.6, or 0.6428...), but still get closer and closer 
to a single number.

But if you start with k = 3.1, you find that the number heads after a 
little while to a pattern where it bounces back and forth between two 
numbers like a pendulum: .55801412 and .76456652.

And if you start with larger values of k, like 3.7, it seems to keep 
jumping around with no discernible pattern: this is chaos. And with 
the starting number of 0.5, after 50 steps you're at 0.921072984, but 
with the starting number of 0.50001, after 50 steps you're at 
0.565549098, a completely different place. This is like the weather; 
this is chaos.

I hope that little introduction helps. You can find a bit more about 
chaos theory in our archives by searching our Ask Dr. Math archives 
for that phrase at:

   http://mathforum.org/mathgrepform.html   

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