You mention $\epsilon$ language, so I'm imagining that the class requires you to apply whatever method they teach. I think such things can often over-complicate these problems. Before you apply specific mathematical tools, I think you should use extremely simple logic to set expectations.
Particularly, why don't you just make a table posing the question of "what does the particle do when moving in X direction?" Let's look at this one for example:
$$x' = f(x) = -x^2$$
please excuse my substitution of left and right for negative and positive respectively
- Displaced right of origin $\rightarrow$ Is moving left
- Displaced left of origin $\rightarrow$ Is moving left
Is this stable? The system will return to $x=0$ in the case that it starts with $x>0$. It is right of origin and moving left. In the other case it is left of origin and moving left, so it is headed for $-\infty$. No, this is not stable.
Rinse and repeat.