I . Introduction

In the 18th 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 This conception has been replaced by the idea of a "cosmic lottery" ever since.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 .

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/(2p-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.... = x0
.....

x3p = 0.110110110110.... = x0
x3p+1 = 0.01001001001001...= x1
x3p+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 xi close to 0.001 (=1/16), a xj close to 0.0001 (=1/32) and so on for i and j, integers. The serie is coming close to 1/2p (p integer) indefinitely.
Theses points 1/2p are called accumulation points. The whole accumaltion points is called strange attractor of the series. Here it is {1/2p, 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.