How does one Linearize and analyze such a system?

Just noticed I could edit this, so from my comment below:

I am trying to get a feel for what analysis us used beyond the introduction I have had. The equations for the double pendulum were derived from the second derivative of the equations for position of each mass and the tension of the mass against the rods. So far the only understanding I have been able to get from it is using numerically generated phase planes and plots motion made in maple.