In the 18^{th} century, scientific thoughts were revolutionized by Newton describing the Universe as "regular as clockwork". Thus we were in a world directed by determinism, that means immutable laws ruled the motion of any particules. This conception has been replaced by the idea of a "cosmic lottery" ever since. Nowadays we are beginning to understand that behavior of systems could not be predictable by sharp laws. In fact, simple laws can generate complex behaviour ; a wellillustrating example is the "clown's hat" function.
II  The "clown's hat" function :
Does it exist a function simpler than this one ? Its graphical
representation is composed of two segments.
This function is defined by :
At point 1/2, values coincide; so the function is continous on [0,1] . It is derivable on the entire interval excepted at the peak of the hat ( x =1/2 ). Coarsely, it increases on [0,1/2[ and decreases on ]1/2,1].
We choose an initial value x_{0} called germ and we calculte the following iterations : f(x0)=x1 , f(f(x0))=f(x1)=x2, ... We are going to describe the behaviour of this system according to the value of the germ .
We can generalize this result : a germ that belong to {1/2^{p}} generates a stabilization at the value 0 after p iterations.
We can generalize this result : a germ that belongs to {1/3.2^{p}} generates a stabilization at the value 2/3 after p iterations.
We can wonder if for any germ series can be stabilized.
The germ x_{0}=1/5 generate a strange behaviour. Actually, after several steps, the series oscillates between two values : 2/5 and 4/5. Here (2/5,4/5) is called a twoorder cycle and the germ 1/5 is a cyclic point. More generally, a germ such as x_{0}=2/(2^{p}1) brings to a porder cycle.
To sum up : the "clown's hat" function can be stabilized at 0 or 2/3, or can generate cycles that the length depends on the germ. But is there for any germ a cycle or a stabilization ?
If following parts, we are going to calculate in base 2 to show
that the answer is no. In base 10, dividing a number by 10 comes down
to move the comma left a row . Dividing by two in the base is the
same thing. Thus for a number lower than 1/2 ( the first figure after
the comma is 0 in binary), the next iteration is given by moving left
a row all the figures after the comma. Moreover if x belongs to [0,1]
and if it is written in binary, 1x is obtained by changing the
parity , i.e. inverting 0 and 1
Example : 1  0.00101... = 0.11010.. . The proof is immediat :
x=0.111111... so 2x=1.11111111111.... then 2xx=1.00...= x. So
0.11111111.... = 1.
Example of iterations for x_{0 } :
x_{0} = 0.110110110110.....
x_{1} = 0.01001001001001...
x_{2} = 0.100100100100.....
x_{3} = 0.110110110110.... = x_{0 }x_{4} = 0.01001001001001...= x_{1}
x_{5} = 0.100100100100..... = x_{2}
x_{6} = 0.110110110110.... = x_{0 .....}x_{3p} = 0.110110110110.... = x_{0 }x_{3p+1} = 0.01001001001001...= x_{1}
x_{3p+2} = 0.100100100100..... = x_{2}
We can see here that to have a cyclic germ, a necessary and sufficient condition is that its script in base 2 is cyclic too. But we know that every rationnal number is cyclic in any bases. Consequently, a germ is cyclic for the "clown's hat function" if it is a rationnal number.
What is it happening for others numbers?
In this part we are going to study unrationnal number written in binary.
1  x_{0} = 0.01 001 0001 00001...
This germ (x_{0}=0,2832651 in decimal ) is composed by an increasing number of 0 between 1.
x_{1} = 0.1001000100001.. close to 0.1 ( = 1/2 in decimal)
x_{2} = 0.11011101110.. close to 1
x_{3} = 0.0100010001.. close to 0.01 (=1/4 in decimal)
x_{4} = 0.1000100001.. close to 0.1
x_{5} = 0.11101110.. close to 1
x_{6} = 0.000100001.. close to 0.001 (=1/8 in decimal)
....
The germ is written with an increasing number of 0 so for the
"clown's hat " function, there is an x_{i }close to 0.001
(=1/16), a x_{j} close to 0.0001 (=1/32) and so on for i and
j, integers. The serie is coming close to 1/2^{p} (p integer)
indefinitely.
Theses points 1/2^{p} are called accumulation points. The
whole accumaltion points is called strange attractor of the series.
Here it is {1/2^{p}, p integer}. If the attractor is
infinite, the series is said turbulent or chaotic. Thus a chaotic
series is neither convergent nor cyclic ( cyclic means that the
attractor is finite).
We can wonder if an other example of infinite strange attractor exists for the "clown's hat function".
2 x0=0.01 10 11 100 101 ...
This number is called Champernowne's number. It is composed of the series of integers written is binary. In decimal it is written : 0,4311203.
By iterating, we can see that this germ generate a series that is never stabilized at a vamue but it is coming close to numbers contained by 0 and 1 infinitely. The strange attractor is the interval [0,1]; As it is infinite, the Champernowne's series is turbulent.
To conclude we can sum up the results in the following table :
Germ x_{0 } 
Behaviour 
{1/2^{p}} , p integer 
stabilization at 0 after p iterations 
{1/3.2^{p}} , p integer 
stabilization at 2/3 after p iterations 
rationnal number 
cyclic 
irrationnal number 
strange attractor composed of the entire accumation points. It is said :

With this simple example, we can see that simple law can generate complex and turbulent behaviour such as hydrodynamics instabilities.
Moreover with this case I learnt how using binary script to do quick iterations without using programs. It is very interresting and usefull to do this kind of logical exercice.