Why does a trajectory take infinite time to reach a critical point? I have to prove that:
For an ODE system $x'=F(x,y),y'=G(x,y)$ where $F,G$ are smooth, any trajectory, which doesn't start at a critical point, cannot reach a critical point in finite time.
I have this counter example. Let $F=0,G=3y^2$. Consider a trajectory $(x(t),y(t))=(0,t^3)$ that starts at $t=-1$. It seems to me that at $t=0$, it will reach a critical point $(0,0)$.


*

*What is wrong with my example?

*How should I actually prove the statement required?


Thanks.

Hint: Assume such a trajectory exists, and consider what this would imply for uniqueness of solutions of the system $x'=-F$, $y'=-G$.
