"According to Polybius, who has the most fully developed version of the cycle, it rotates through the three basic forms of government, democracy, aristocracy, and monarchy"

The version you describe is the Anacyclosis, which is equally valid IMO but not what I linked.

"According to Polybius, who has the most fully developed version of the cycle, it rotates through the three basic forms of government, democracy, aristocracy, and monarchy and the three degenerate forms of each of these governments ochlocracy, oligarchy, and tyranny. Originally society is in anarchy but the strongest figure emerges and sets up a monarchy. The monarch's descendants, who because of their family's power lack virtue, become despots and the monarchy degenerates into a tyranny. Because of the excesses of the ruler the tyranny is overthrown by the leading citizens of the state who set up an aristocracy. They too quickly forget about virtue and the state becomes an oligarchy. These oligarchs are overthrown by the people who set up a democracy. Democracy soon becomes corrupt and degenerates into mob rule, beginning the cycle anew."