Excuse me if I am wrong.
It sounds to me like a wonky bulb in there. It starts off making some OK contact, then as it twists, so does the element, eventually slipping in and out of position as it heats and twists around. Then it cools and starts the process over again.
Then again, I could see just basic wiring issues as well. The bulbs should be fairly easy to troubleshoot.
The lack of other indicators leads me to the bulb. After seeing a par64 can bulb break and having to replace it, I was thinking "that's a nasty looking element". The bulb blew when it was run into by accident by a wrestler, so with the heat it definately weakened the filament and made it break. Also, I have yet to blow a bulb in a Par38, but I've broken a few bulbs with the filament still intact(odd!). Anyhow, noting the twisted nature of many bulb filaments(to maximize how much is excited and produces light), this idea seems plausible.