For a real function $f$ on $\mathbb R^n$, such that no trajectories of the gradient escape to infinity, what are necessary and/or sufficient conditions so that each trajectory limits to a unique point of the critical set?
I can produce a function $f$ on the unit disk $D$ in $\mathbb R^2$ with the property that $\partial D$ is in the critical set and each point $x\in\partial D$ is a limit point for any generic trajectory. I would like to see why exactly this function fails to have unique trajectory limits, in terms of some property such as Lipschitz continuity (which I believe isn't sufficient) or Hölder continuity. My guess is that it is sufficient if the gradient, and not the function, is Lipschitz continuous.
This seems like an elementary question, but I see it's related to Hilbert's 16th problem, and possibly it's more subtle than I understand.