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 of motion made in maple.

I have seen an overview of the analysis mentioned on wolfram's site and am reading into that, but these books on mechanics are fairly broad, and I am looking to see if there is a specific field I need to look into.

Edit in response to comment: Not looking for this to be simplified to the level of a pendulum, I just need to know how to approach the problem. I do not know what analytical tools would be used to better understand the system(as I understand it, there is no solving for such a complicated system in the same form that a regular nonlinear pendulum can be).