index

Abstract

I.Historical and physical backrounds

II.Analisis of the Lorenz's dynamic system: equilibrium and stability

III.Numerical simulation:
- observation 3D of the chaotic comportment of the system
- visualization of the pitchwork bifurcation

Conclusion & bibliography


 




II. Analisis of the Lorenz's dynamic system : equilibrium and stability
 

II.1 Lorenz's system

This system, in its reduced form, is written as
 

(S) =     where   .  It can also be presented as      with .


II.2 Equilibrium
 

The equilibriums of the Lorenz's system exist for    that is to say  .

The solutions of this system are:

        if  a=0 then x=y=z=0 is solution and if   solutions are  .


 
 

II.3 Stability
 

By applying a little pertubation to the system in its equilibrium position, we can realize a limited development.

Then 

The system is simplified and linearised by ignoring the quadratic term.

   (1)
 

It is clear that the Jacob matrix () sign will  light the points stabilities. This supposed to study the proper value of this matrix..
 

a) Stability of 
Placing Xe=(0,0,0) in the Jacob form, the matrix becomes ,

which caracteristic equation is =0

-b is clearly a proper value. Noticing that the second degree polynom discriminant is positive, the two other roots are then real.

Their sum S and product P correspond to  .

One can now deduce that:

- for r<1, all the proper values are negative : the system is stable for Xe=0
- for r>1, P <0 meaning that one of the second degree polynom roots is negative. The system is unstable for Xe=0.
 

Informations about the trjectories

The small pertubation is expressed in the base of the proper vectors associated to the proper values found before.
 


 

In this base, (1) turns into 

The system to solve is now :   which solution is 
 

This solution shows that the trajectories converge only if the proper values are negative. It is the case for r<1.
 
 
 
 
 
 
 

b) Stability of 

These stabilities required a quite long parametric study of b, r and s. We will not deatil it. Fixing b=8/3 and s=10,
stabilities have different form according to the r value . For r<24.7, the form can be visualized and corresponds to a pitchwork bifurcation.
We can also obseved the system symetry.


 

c) Conclusion