I don't think it's a given that revolution and violence reset the cycle to liberalism. There would appear to be plenty of examples where the chaos just ushers in a greater degree of authoritarianism and how much liberalism that follows a power change is just caused by the period of new power consolidating its control?

Even if it's a matter of picking the right start and stop times to a cycle, every iteration would be heterogeneous enough that calling it a cycle is probably missing some important behaviors.

Fortunately -in many civilized countries- we have institutionalized and ritualized the act of having a revolution every 4 years.

As with many such rituals, -and as long as everyone sticks to the prescribed acts- this reduces the bloodshed immeasurably. With a little luck, the only victims are dead trees and broken pencils.

We're on a loop alright, but it's caused by the interplay of the more numerous working/underclasses and elites struggling to stay on top.

Currently we are at the bit of the loop where the underclasses are thrashing out and the elites are trying to channel that rage somehow.