Presentation of common cases

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:




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






The mane of this instability is fork.


This diagram are the Hopf bifurcation.




 Few aspects of the instability:


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(U2)
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.

A equilibrium is sable by tree ways:

Moreover, we could assembly the tree models.