Associated Topics || Dr. Math Home || Search Dr. Math

Chaos Theory and Converging Sequences

```
Date: 2/14/96 at 10:32:10
From: sarah safraz
Subject: sequences

Hi. I'm emailing from London.  I'm a 28-year-old having to resit a
math exam!  As part of my course work I'm investigating a process
of dividing and adding (i.e. take a number, divide by 5, add 2,
write down result, repeat process using the result).  I realise
that using this particular process my sequence will converge
towards 2.5, where nothing else can be done to it.  However, by
reversing the process, i.e. adding 2 first then dividing by 5, my
sequence will diverge to what I guess will be infinity.  Trouble
this really be all there is to do?  Could I do anything to jazz it
up a bit?

Yours in frustration,

Billy (Sarah) Burlingham
```

```
Date: 2/14/96 at 11:17:4
From: Doctor Byron
Subject: Re: sequences

Wow!  What a great observation!  You may not realize it, but the
study of sequences like the one you just mentioned has become one
of the hottest topics in mathematics in recent years.
Mathematicians have found sequences like this that diverge,
converge, jump back and forth between different numbers, and even
oscillate in wild chaotic patterns.  Speaking of chaos, you may
have heard the term 'chaos theory' before (perhaps the movie
Jurassic Park?).  One of the more famous examples of chaos theory
is a sequence like the one you just described.

Choose some constant, C, and an initial value x1.  Now create a
new number x2 according to the equation:

x2 = C(x1)(1-x1)  [This is called the "logistic equation"]

Depending on your choice of C this equation can have wildly
different effects.In most cases, the behavior of the system is not
dependent on the initial choice of x1 (as you noted in your case
with x/5 + 2).  For certain values of C, however, the behavior of
the system in wild and chaotic (leading to some great computer
graphics, I might add).  Actually, the chaotic behavior starts
somewhere around C > 3.57.  For C = 3.0 you get a system that will
eventually oscillate between four states.  If you are interested
in learning more about Chaos Theory, there is a book called
"Chaos" by James Gleik which is very readable and manages to
discuss the subject without too much heavy mathematics.

To get back to your question of why x2 = (x1)/5 + 2 converges to
2.5, however, this sequence can be rewritten in a
generalexpression of the form:

xn = [(.2)^n]x1 + [2 + (.2)2 + (.2)^2*2 + (.2)^3*2 ... + (.2)^n*2]

The first expression, which depends on x1, gets smaller and
smaller as n gets larger until it is essentially insignificant.
You will notice,however that the second long expression in terms
of 2 and powers of 0.2 does not depend on x1.  It is possible to
show that this series has an sum which gets closer and closer to
2.5 as n approaches infinity.  This is why you observed the
behavior you did.

Good luck with your math exam.  This is a very rich topic, and I
hopeyou enjoy exploring it!

-Doctor Byron,  The Math Forum

```
Associated Topics:
College Analysis

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search