I have been re-engaging with a bit of theory.

When we find any unmistakably periodic behavior from an organism, one sensible theory is that something is oscillating to control that behavior. (Bob Port)

I need to consider why this is important. What is it that is oscillating? The undamped oscillator of task dynamics is a hack. That’s not a good enough description. But the maths gets hairy when you get more complex.

First, since the theory specifies attractors in terms of phase angle,
we expect that at least for moderate changes in rate (that is, changes
in the duration of the repetition cycle), the attractors should be
unaffected in terms of phase but vary in direct proportion with cycle

My stance here would be to shun the notion of control, and recognize that periodic system behavior is simply a  common form of viable, stable self-organized behavior.  It needs no controller, and indeed, it makes no sense if there is a controller.  It only makes sense because this is how simple systems, each with some autonomy, will couple.