Condition For Existence Of Phase Flows I am a bit confused about the existence of one-parameter groups of diffeomorphisms/phase flows for various types of ODE's. Specifically, there is a problem in V.I. Arnold's Classical Mechanics text that asks to prove that a positive potential energy guarantees a phase flow, and also one asking to prove that $U(x) = -x^4$ does not define a phase flow- and these having me thinking.
Consider the following two (systems of) differential equations:
$\dot x(t) = y(t)$, $\dot y(t) = 4x(t)^3$ 
and
$\dot a(t) = b(t)$, $\dot b(t) = -4a(t)^3$.
Both phase flows might, as far as I see it, have issues with the fact that the functions $\dot y(t)$ and $\dot b(t)$ have inverses which are not $C^\infty$ everywhere. However, the $(x,y)$ phase flow has an additional, apparently (according to Arnold's ODE text) more important issue- it approaches infinity in a finite time.
Why, though, do I care about the solutions "blowing up" more than I care about the vector fields' differentiability issues? 
$\textbf{What is, actually, the criterion for the existence of a phase flow, given a (sufficiently differentiable) vector field?}$

How does Arnold define "phase flow"?  As far as I know, part of the definition of a flow requires the solutions to exist for all $t > 0$.  If they go to $\infty$ in a finite time, they don't exist after that.  On the other hand, I don't see why not having a $C^\infty$ inverse would be an issue.  
