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Possible that your guide star has another star very very close to it that you may not be able to see with your eyes.
I had this exact issue when I was using Alnitak to guide with when imaging the Horsehead Nebula. There is another very small star right next to it. Chose a different guide star and problem was gone...
However that does not explain your cyclical issue... perhaps atmospheric disturbance . Don't know...
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