I.Historical and physical backrounds
II.Analisis of the Lorenz's dynamic system: equilibrium and stability
- observation 3D of the chaotic comportment of the system
- visualization of the pitchwork bifurcation
Conclusion & bibliography
II.1 Lorenz's system
This system, in its reduced form, is written as
(S) = where . It can also be presented as with .
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 .
By applying a little pertubation to the system in its equilibrium position, we can realize a limited development.
The system is simplified and linearised by ignoring the quadratic term.
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.