Since I have to do things with pictures in order to properly visualize what's going on in a setup like the pendulum, I thought I'd put this in. This could be completely wrong. It's difficult to chase down information on this sort of thing and I don't have a galvanometer lying around to actually do empirical testing, so YMMV.
At any rate, what I have pictured is a pendulum swing going in one direction and inducing a current. Now from what I understand, the direction of the magnet going past the coil determines the direction of current flow. Which means that when the magnet goes the opposite direction, the current flow and battery diagrams are reversed.
Which means that whatever is in the black box has to be able to deal with current going both ways in the trigger wire (the same thing happens in the power wire, but probably does nothing)--and could be the method by which the batteries are flip-flopped.
The actual electromagnet portion (power wire) would not have to care about which way the current is flowing in the trigger, since all it is concerned with is turning the power on long enough to 1), give the incoming magnet a kick in the direction it's going and 2), generate the radiant spike. A clue to this portion of the circuit is that the higher the impedance on the charging end, the higher it swings and thus the stronger the kick of the electromagnet. This means that there's a feedback mechanism at work on that part of the circuit.
That's all the ideas I have for now, feel free to pick it apart/point out inaccuracies.