In nonlinear dynamics of biology, we would treat the homeostatic point as a "stable focus" and the threshold beyond which the system fails us would be an "unstable limit cycle". It is basically a dot in the middle of a circle. The dot attracts solutions towards it, the circle pushes solutions away from it (so solutions inside the circle get smaller, solutions outside get bigger). Imagining the potential energy analogy, it would mean that you basically have a huge bowl inside of a mountain with a ball in it. Small perturbations of the ball inside the bowl, and it just comes back to the bottom of the bowl, but a sufficient perturbation and you knock the ball over the walls of the bowl and it flies faster and faster down the mountain outside, until it reaches the bottom, at a much lower state than the bottom of the bowl (having significant biological effects in the whole organism).
There's actually a slight difference with the mountain analogy. Solutions don't have to swirl around as the approach the point. In biological systems, we are generallly plotting two variables against each other that do make a deterministic cycle, so the motion will be circular like in the flow diagram.