From: ba@sxb.bsf.alcatel.fr (@Bertrand AUGST)
Newsgroups: sci.math
Subject: Re: cos and sin
Date: 3 Sep 1996 10:29:15 GMT

> Such a sequence does not necessarily have 
> to converge. For example, when f(x) = x^2, then the sequence converges to 0 
> when |a_1| < 1, it converges to 1 if a_1=1 or a_1 = -1, and otherwise it does 
> not converge at all. For the sine and cosine functions, it does converge 
> because (informally speaking) they do not increase or decrease quickly.

You can even have stranger behaviours with fixed points :
Consider the function p(x)=k(1-x)*x, with x constant.

Concretely, you can see this p as the function that gives a population p(x) at
generation n+1 depending on the population x at generation n.
k gives a rate of fertility for this population.
To simplify the problem, we work with 0<x<
The bigger k is, the faster population will grow from a generation to the next.
The bigger x is (aka the more parents their is),
the bigger p(x) will be if (aka the more children their will be), provided that
their is no overpopulation (1-x)

Now, return to maths. We want to know the limit population after many generations,
depending on k (fertility) and initial population p0.
So, what we have to do is to calculate :
L=p(p(...(p(p0))))
We have a fixed point problem : p(L)=L
We remark that p0 has no influence on L (exepted if p0=0!)

We could expect to have such a graph :
L
|                           
|                 .         .
|        .   
|    .
|  . 
| .
|.
+-----------------------------> k

But in fact, what appears is that p(x) is non-linear, and
that its behaviour is chaotic.
for big values of k, population doesn't converge on a single
value L, but oscillates between many values :
- first : 2 values
as you increase k
- 4 values
- 8 values
...
infinity of values

That's what we call doublings periods.

L              /--+++++++++
|           -- \__+++++++++
|         /         ++++++
|       + \         ++++CHAOS
|     /     -- /--++++++++
|   /   ^      \__++++++++
| /     |
|/      |
+------ | ---------------------> k
0       |          3
    bifurcation

The first man who has discovered this a few years ago is Feigenbaum.
That's why this diagram is called a Feigenbaum diagram.
Try to trace it on your computer (it's quite easy to do), it will
be undoubtedly much more beautiful than my text diagram.

This phenomenon appears concretly in many domains :
- population of some fertile animals (rabbits, crickets) or viruses (measles)
  that become unpredictable from one year to the other
- fluid mechanics (turbulences)
- meteorology
in every phenomenons that can be put in equations with a p(x)=k(1-x)x,
even after many simplifications.

Indeed, this appears even if you complicate the equation for p(x), provided
that the "essence" of the equation is kept.
ex: p(x)=sin(x)(1-sin(x))*k must work

Moreover, independently of which equation you take, you can see a general
constant appear :
i=0
Take k=0, and let it slowly grow
Each time a bifurcation appears, memorise the value k[i]=k, and increase i

At the end, consider the ratio k[i]/k[i+1] when i->infinity.
It will converge to the feigenbaum constant (4.669...).

This constant is always the same (you can modify the equation for p(x), you
can change initial population). In fact, it is as general as pi or e, and
appears in many chaotics phenomenons. You can even see it in the mandelbrot set.

Strange, isn't it ?

On this subject, I'd have a question : Does anyone know if their is a way to
calculate this feigenbaum constant in such a way :
K=SUM(i=0 to infinity, Ui), where a general term for Ui is known.
(as we can do for e or pi)

PS : Sorry for my english.

Bertrand AUGST. ba@sxb.bsf.alcatel.fr