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
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.