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II-EQUILIBRIUMS

We have the equation: (1)
(1)
So, the equilibriums of this equation (1) result from the equation (2):
(2)

1- Equilibriums and stability of (2)

We reconize a dynamic system with a potentiel. So, we know that the equilibriums of (2) are:

(P,V'(ue))=(0,0). Futhermore, we know that stability of this equilibres result from the sign of V''(ue).

So, we are going to study V'(u) for differents values of the parameters.

• Graph of V' for >0 and >0:

If (>0 or ) we have one stable equilibrium  because V''(ue)>0.

If (<0 and ) we have 2 stable equilibriums (ue1,ue3) because V''(ue1)>0 and V''(ue3)>0 and one  instable equilibrium ue2 because V''(ue2)<0.

• Graph of V' for >0 and <0:

If (>0 or ) we have one instable equilibrium  because V''(ue)<0.

If (<0 and ) we have 2 instable equilibriums (ue1,ue3) because V''(ue1)<0 and V''(ue3)<0 and one  stable equilibrium ue2 because V''(ue2)>0.

Rem : We don't plot V' for (<0 and <0) and (<0 and >0) because we find the same graphs with a symetry of the axe u.
2-Graph of equation (2) in the plane (u,P)

We considere the case = -2/27, =1, =1, K=0,06 .

So, we have one instable equilibrium ue2= -1/3 and 2 stable equilibriums                            ue1= -1/3-(1/3)^1/2  and  ue3= -1/3+(1/3)^1/2.

With Matlab, we obtain the 3 differents curves for each equilibrium.

• Graph for ue3:
• Graph for ue1:
• Graph for ue2:

These curves corespond very well with the analytic theory of dynamic system with a potentiel.