**Results**

**Our study is about the reaction of the pendulum
when we modify the frequency of the density variations.**

**In the first case, the frequency is the double
of the system frequency so, when the two variations are in phase, the movement
is the same than a swing. So, we must observe an entertained movement for
the little values of the gravity coefficient.**

**In the second case, we make tests with a frequency
of the gravity variations equal to the frequency of the system.**

**In all the simulations, the average gravity
is taken equal to 9, the mass is 1Kg and the rope is one meter long, so
the pulsation of the system is 3rd/s.**

__1) The swing mouvement.__

**In this first case, the pulsation of the gravity
(6rd/s) is twice the pulsation of the system (3rd/s) and gravity coefficient
is 0,3.**

**After a short period, the system converge to
a stable solution.**

*variation of the angle with the time*

*variation of the angle with the velocity*

**This first case is the movement expected.**

**If we increase the coefficient of gravity (
0.9 ), we obtaine the results following.**

**There seem to be different attractors, but
finaly the system converge to the same kind of solution than in the first
case.**

**Now, we take a coefficient of gravity equal
to 1,1. So in this case the gravity is oscillating between -0,1g0
and
2,1g0.**

**We get the results following :**

**In this case, the system diverges.**

**If we continue to increase the coefficient
of gravity, we get curious results. In the following case the coefficient
is equal to 10, it has no physic sens.**

**The solution we get seems to be chaotique.**

__2) Equality of the pulsations.__

**Now, the two pulsations are the same.**

**If we take a coefficient of gravity equal to
0.2, we get the results following :**

**That result represents a system amorted.**

**If we continue to increase it to 0.6, we get
the results following :**

**The system is converging.**

**If we still increase the coefficient (k=1.5),
we get :**

**It seems to be chaotic with no attractor.**

**Now, we take k=2**

**The attractor is quite strange.**