I will present you in this pages some simple examples of dynamic systems. A dynamic system is a system of function where :

First example : node saddle bifurcation

The phase space is equal to 1

The control space is equal to 2

To study this system, it must be discus of its equilibrium.

It is plotted here the function

For the different values of the parameters we have:

trajectories

For and for example we have this scheme

It is notable that a trajectory can’'t cross itself, so, a node is composed by two trajectories. The possible trajectories are :

The point x1,

the point x2,

]x1,x2[,

]- oo , x2 [ ,

]x1, +oo[

Stability of the different equilibrium

To study the stability of the different equilibrium, it is disturb : x(t)= x1 + u(t)

So, we have to cases

stable

unstable

If we come back to the function parameters we can easily see that one part of the curve is stable and the other is unstable:

Second example : Fork bifurcation

The phase space is equal to 1

The control space is equal to 2

To study this system, it must be discus of its equilibrium.

It is plotted here the function

For the different values of the parameters we have:

Stability of the different Equilibrium

Plotting the derived of the function f it is easy to find the stable and unstable equilibrium of the function

Third example : Hopf bifurcation

The phase space is equal to 2

The control space is equal to 4

The calculation is made in the polar space

All calculation made, it can be draw to cases

**first case : **

In a 3-D space:

It's the supercritical Hopf bifurcation

**Second case : **

In a 3-D space:

Undercritical Hopf bifurcation