Henri Poincaré feared to say that the solar system was not exactly periodic and could be unstable (in the traditional use of "PIO" divergence or non convergent oscillation).We know that periodic dynamic is governed both by phase and frequency, but traditionally lack of math tools focused on the only phase control which does not involve energy use at the contrary of frequence modification.
Another well known oscillation in airliners is passengers' hyperventilation. Frequense of breath is increased by different factors disturbing the CO2 and O2 regulation. Trying to correct it with phase has no chance of success.
Sometimes periodic movements are not more or less sinusoidal oscillations, I.e. chalk squeaking on the blackboard when pushed and doing rapid successions of jumps.
Progressively theory of chaos taught us that difficult art of controlling oscillations.
Multi variable problems are more difficult to describe in short words.
"Pilot" in PIO may be Autopilot too, as we see it in the KC-135R dutch roll report when the captain tried to stop it with A/P. The notion of "fault" is of no use in science. We are concerned by cause.