# 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
are given by solving the equation:

The two solutions are:

### Stability of the equilibrium
points

We could write:* x(t)=x*_{i}+u(t)

where *x*_{i} is an equilibrium point and *u(t)*
is very small is front of *x*_{i}.

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 *x*_{i }is stable or
unstable (remember that* x*_{1}>0 and* x*_{2}<0):

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

*alpha<0*

*alpha>0*