If the universe is cyclic, then that would define it as a closed loop without any energy being removed or added. The very first instance of yourself traveling through time would break the loop by removing themselves from their iteration of the universe and reinsert into a future iteration. During those two points, the universe would now function with a deficit. This deficit could affect how it cycles to subsequent iterations.

What if that discrepancy in energy affects when the universe starts to contract or it's speed to the big crunch/bang and subsequently the time to, and speed of expansion. Maybe it could even prevent the big crunch from reaching critical mass, where it would normally trigger a big bang, and stop the cycle altogether.