I'm going to be lazy and skip writing a big long introduction or explanation for this. The Wikipedia article on chaos thoery is pretty good and explains how chaotic systems are sensitive to initial conditions. The mathematics are too complex to accurately reproduce here, but the links below can show them correctly:
http://www.chaoticpendulums.com/chaos-theory-a9.html (simple explanation of chaos theory)
Here's a neat version made from two square plates:
You can buy a double pendulum from chaoticpendulums.com, but it's more fun to build your own. Look at the pictures, look at the CAD files, watch the video, and then go make one.
Support Amazon.com for sponsoring this science fair contest, buy ball bearings online.
Standard shop and power tool warnings apply, but I have to provide a warning specific to the double pendulum. The bottom pendulum can get moving very fast and because it's chaotic, it's unpredictable. If your hand or face is in the wrong place and the wrong time, you can get seriously hurt. The best thing to do is to set it in motion and then stay out of the plane of rotation.
I fixed the video! Sorry about that. It's viewable in step 10.