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 
is, I don't know what else to say about this investigation!!  Can 
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