**The pendulum in the absorbed
oscillations case**

__Equations__

The movement can be modified by adding a friction. It
correspond to a more natural movement. The pendulum is oscillating during
a certain time (depending on the friction) and always finish in an equilibrium
position.

We have choosen to use a friction term proportionnal
to the speed of the pendulum, with a friction coefficient, lamda. It gives
us the following equation:

and the system:

__Results__

There is different results depending on the value given to the parameter mu. Pathlines in the phase portrait converge to a zero value.

The spirale in the phase portrait correspond to a pendulum
that oscillate and slow down in the position of equilibrium.

If the pendulum has a initial speed that makes him complete
one or several rotation, it will converge to a value different from zero
but in a multiple of 2*Pi that are the zeros of the system (positions of
equilibrium). We can see it clearly on the time evolution of the angle.

It correspond to the different attractors (wells) in
the domain.

When the coefficient mu is very important, there are not many oscillations around zero.

Here we can see an attracting valley around zero. All
the pathlines reach zero with almost the same direction. It must be an
eigenvalue of the system in this precise configuration.

On the temporal evolution, we can see that there is no
more oscillations.