In this paragraph, we are going to present you the common case of instability
and a few aspects of the instability theory.

For each cases, we'll plot the bifurcation diagram and give some comments.

The previous example:

*alpha<0*

*alpha>0*

This case is the simplest of the common cases present here. Its name is node-col.

*alpha>0*

*alpha<0*

The mane of this instability is fork.

This diagram are the Hopf bifurcation.

*Alpha.r<0*

*Alpha.r>0*

Few aspects of the instability:

- Stability of the equilibrium:

Let DX/dt=F(X) be our case.

X0 is an equilibrium point. We could write: X=X0+U

So dU/dt=F(X0+U)=F(X0)+D.F(X0).U+o(U^{2})

But F(X0)=0, so **dU/dt= D.F(X0).U**

So the stability of the equilibrium point depend on the sign of the eigen values. Let s be an eigen value.

**Re(s)<0 -> X0 is stable**

**Re(s)>0 -> X0 is unstable**

If Re(s)=0 the case is ambiguous.

- Theorem of the dissipative system

A equilibrium is sable by tree ways:

- Node-col (steady)
- Fork (steady)
- Hopf (unsteady)

Moreover, we could assembly the tree models.