Did Romans recognise the fall of the republic?

[Epictetus]

Today we recognise the “Fall of the Republic” as a decisive event. Conventionally, we see a fundamental change in governance taking place. We also see a change in public policy and strategy, to some extent. We even use terms that reflect this, such as the “Republican period” or “Imperial period.”

So here’s my question: at the time, or shortly thereafter, did the Romans think that a decisive shift in the res publica had taken place?