General study of a simple case

 

This first paragraph aims at present the methods to study a dynamic system.

The case, we have chosen, is:

We are going to study:

The dimension of the phasus space and the control space

The equilibrium points

The stability of the equilibrium points

The bifurcation diagram


The dimention of the phasus space and the control space

The phasus space is defined by R2 ans the control space is defined by R. Indeed the definition of the dynamic system is:

The equilibrium points

The equilibrium points are given by solving the equation:

The two solutions are:

 

Stability of the equilibrium points

We could write: x(t)=xi+u(t)
where xi is an equilibrium point and u(t) is very small is front of xi.

So we have:

where is null because it's an equilibrium point.

Then:

So by replacing this expression in the equation of the dynamic system, we have:

By using a Taylor developement (u(t) ­>0), we have:

We have to solve a first order differential equation. The solution is:

So the stability of the equilibrium is depending on the sign of a.

For alpha<0 the point is stable, and for alpha>0 the point is unstable.

Let's look at this table to find if xi is stable or unstable (remember that x1>0 and x2<0):

 

Bifurcation diagram

The continuous line represent the stable point, and the dash points the instability.

alpha<0

 

alpha>0