Induced Phugoid Oscillation?

It takes another engineer to understand where you are coming from, so yes, the concepts you descibe are familiar to me, but alien to most reading this thread. The basic control system breaks down due to diabetes, and we rely mainly on manual intervention to correct it. The sample rate is usually very low, and there is significant lag response so that small pertubations can lead to underdamped oscillation such as you describe. It is an open loop system with very little feedback, so it is difficult to be predictive. But with experience it is possible to introduce some predictive actions that lead to overdamped responses in time. It is probably impossible to get a critically damped response.

The introduction of CGM monitoring will help, but until we get a true closed loop system it will always be hit and miss, Some newer T2D medications do self regulate and have self limiting control ranges.

I suppose the nearest thing you may find in the literature would be "chasing sugars" though even that is not entirely there and I have heard and seen it used to describe subtly different situations.

I think Oldvatr pretty much nails the key issues in that the typical sample set is so sparse as to be difficult to make really good decisions. Coupled with the variable lag that you can get from the "fast" acting insulin this further complicates the control inputs available to a person.

All that being said I think this is potentially a really interesting area of research since I think it could certainly provide some really useful data for the algorithms required to effectively run a closed loop artificial pancreas system.