Theoritical study

Let us remind you that the model is : 

1/ Fixed points :

2/ Stability analysis :

3/ Rate of area contraction (dissipation) :


1/ Fixed points :

With discret models, a fixed point can be defined by Xk+1=Xk and Yk+1=Yk, which leads to 

and then .

The resolution of the first equation implies that there can be fixed points only if .

That is to say .

It is then obvious that if a>0, there are always two distincts fixed points :



2/ Stability analysis :

We now have to determine wether these points are stable or not. Therefore, let's re-write the problem under the following form : 

The sign of the eigen values on the matrix  will determine the stability of the fixed points.

It is easy to find that the eigen values are : 

A complete general study of the stability could be a bit too long therefore, we have decided to calculate the numericl values of l1 and l2 in the case a=1.4 and b=0.3 at the fixed points calculated in the former paragraph. Therefore, when you will run the "fixed points" version of the program, eigen values will be printed. If one of these values is negative, then the point is instable.


3/ Rate of area contraction (dissipation) :

Cahotic systems are always affected by sensibility to initial conditions (as will be discussed in the interpretation of the numerical ressults), and the loss of memory concerning initial condition. To respect the second characteristic, it is necessary that areas are contracted. The rate of contraction of the function F is represented by its Jacobian . Consequently, areas are on average multiplicated by |b| between each iteration. If |b|<1, areas are contracted (b=03 in the case of Henon attractor).


I suggest you now have a look at the numerical results but if you prefer to go back to the main page, I don't mind!