Chaos Theory
Consider a population of organisms for which there is a
constant supply of food and limited space, and no
predators.
Many insect populations in the temperate zones fit this
description at certain times in their history.
In order to model the populations in successive
generations, let N  denote the population of the nth
generation, and adjust the numbers so that the capacity of
the environment equals to 1, which means that
0  N  1 .
One formula, called logistic equation, has gained
widespread fame:
N n 1  N n (1  N n )
for
0  N  1
where the parameter  indicates the rate of which the
populations grow.
y  Q ( x)  x(1  x)
y

4
12
1.0
0.8
0.6
x
0.4
0.2
0.0
1.0
1.5
2.0
2.5
c
3.0
3.5
4.0
x
1.0
0.8
0.6
0.4
0.2
0.0
3.82
3.83
3.84
3.85
In general,
k
If  n1     n , then Q  has a 2 -cycle for
k  0,1,2,....., n
It is known, but difficult to prove, that  k 1  1  3   k

for k  2,3,....., and that the sequence  k k 1 has a
limit   given by
  3.61547   
3.86
3.87
3.8
The number   is sometimes called the Feigenbaum
number for the quadric family, named after the physicist
Mitchell Feigenbaum, who in the mid-1970,s had found a
very precise value for it.
Let
dk 
 k   k 1
 k 1   k ,
for
k  2,3,4.....
Feigenbaum found that the sequence d k k 1 converges
to a number we will denote d  , where
d   4.669202   
What is astonishing is that this constant d  seems to be
universal. It is referred to as Feigenbaum constant,
because Feigenbaum was the first to discover it and its
universality.
2
Exercise f  ( x)  x  1 4  
Period-doubling bifurcation
Pitchfork bifurcation