

Parametric resonance is a resonance phenomenon different from normal resonance and superharmonic resonance because it is an instability phenomenon.
The vertically driven pendulum is the only driven pendulum in the lab which has the
same stationary solutions as the undriven pendulum, namely
In order to investigate the stability of a fixed point, you have
to linearize the equation of motion around a fixed point.
For
Even though the Mathieu equation is a
linear differential equation, it can not be solved analytically in terms of standard functions. The reason is that
one of the coefficients isn't constant but timedependent. Fortunatly, the coefficient is periodic in time.
This allows to apply the Floquet theorem. It says that in a linear differential equation or a system of
linear differential equations there exists a set of fundamental solutions (from which one can build all
other solutions) where all solutions can be written in the form is fulfilled, where n is an integer defining the order of parametric resonance. In case of damping, a driving amplitude a exceeding a critical value is necessary for destabilization. 
There is a simple intuitive understanding of the parametric resonance condition. And maybe you already have such intuitive knowledge! Imagine you are on a fair and you want to swing a swing boat (did you ever try?). Standing in the boat you have to go down with your body when reaching a maximum because you want to speed the boat up by putting an additional acceleration to the acceleration of gravity. If you do that near the forward and the backward maximum of oscillation, you are just realizing firstorder parametric resonance because you are moving periodically up and down with a frequency just twice the frequency of the swing boat. If you go down only every nth maximum you will have parametric resonance of order n.
The onset of firstorder parametric resonance can be approximated
analytically very well by the ansatz:
(3) 

QUESTIONS worth to think about: 
