NOTES
Now, initial conditions matter a lot. I've started the system with given initial angular velocity for the lower pendulum, and compared the result with the performance that occurs when everything is completely the same, except that the initial angular velocity is slightly different (multiplied by .9999, or .99999 or so on). The plots show angular position for one of the pendulums - and as you can see, after less than a minute, there are obviously growing difference between the results obtained from slightly different starting points.
One question raised during the talk was whether these differences were transient. The baseline simulation was given about 2.5 minutes to dissipate transients prior to the tiny disturbances that were introduced at time zero (above) - this suggests that transients are no longer an issue.