I . Introduction

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 This conception has been replaced by the idea of a "cosmic lottery" ever since.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 well-illustrating 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].

III - Iterations - Germ

We choose an initial value x₀ 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 .

• x₀ = 1/2 : x₁=1 , x₂=0 , x₃=0 , ....There is stabilization at the value 0.
• x₀ = 1/4 : x₁=1/2 , x₂=1 , x₃=0 , x₄=0 , ....There is the same stabilisation.

  We can generalize this result : a germ that belong to {1/2^p} generates a stabilization at the value 0 after p iterations.

• x₀ = 1/3 : x₁=2/3 , x₂=2/3 , x₃=2/3 , ....There is stabilization at the value 2/3.
• x₀ = 5/6 : x₁=1/3 , x₂=2/3 , x₃=2/3 , ....There is the same stabilisation.

  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.

   

   

IV - Cycles

We can wonder if for any germ series can be stabilized.

The germ x₀=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 two-order cycle and the germ 1/5 is a cyclic point. More generally, a germ such as x₀=2/(2^p-1) brings to a p-order 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, 1-x 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 2x-x=1.00...= x. So 0.11111111.... = 1.

Example of iterations for x₀ :

x₀ = 0.110110110110.....
x₁ = 0.01001001001001...
x₂ = 0.100100100100.....
x₃ = 0.110110110110.... = x₀
x₄ = 0.01001001001001...= x₁
x₅ = 0.100100100100..... = x₂
x₆ = 0.110110110110.... = x_0
.....

x_3p = 0.110110110110.... = x₀
x_3p+1 = 0.01001001001001...= x₁
x_3p+2 = 0.100100100100..... = x₂

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?

V - Stange attractors

In this part we are going to study unrationnal number written in binary.

1 - x₀ = 0.01 001 0001 00001...

This germ (x₀=0,2832651 in decimal ) is composed by an increasing number of 0 between 1.

x₁ = 0.1001000100001.. close to 0.1 ( = 1/2 in decimal)
x₂ = 0.11011101110.. close to 1
x₃ = 0.0100010001.. close to 0.01 (=1/4 in decimal)
x₄ = 0.1000100001.. close to 0.1
x₅ = 0.11101110.. close to 1
x₆ = 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.

VI - Conclusion

To conclude we can sum up the results in the following table :

Germ x0	Behaviour
{1/2p} , p integer	stabilization at 0 after p iterations
{1/3.2p} , 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 : chaotic if the number of accumation points is infinite calm on the contrary

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.