__The
parametric oscillator : the simple pendulum without friction in a various
gravitational range.__

In fact, this is an alternativ movement of the point O that reproduces such conditions.

We have a new expression for the gravity :

The expression of the dynamic equation is now:

This equation is not integrable in the case of an ordinary law for g(t).

Moreover the origin point is not necessary stable...

We take g periodical (Hill's equation) with b circular (Mathieu's equation):

So the equation becomes:

The solutions can be written under the form below:

Stability of the solutions:

For m
small, and w» w_{o}_{.}

_{We obtain this diagram of stability:}