  We consider a class of ordinary differential equations in $d$-dimensions
featuring a non-Lipschitz singularity at the origin. Solutions of such systems
exist globally and are unique up until the first time they hit the origin, $t =
t_b$, which we term `blowup'. However, infinitely many solutions may exist for
longer times. To study continuation past blowup, we introduce physically
motivated regularizations: they consist of smoothing the vector field in a
$\nu$--ball around the origin and then removing the regularization in the limit
$\nu\to 0$. We show that this limit can be understood using a certain
autonomous dynamical system obtained by a solution-dependent renormalization
procedure. This procedure maps the pre-blowup dynamics, $t < t_b$, to the
solution ending at infinitely large renormalized time. In particular, the
asymptotic behavior as $t \nearrow t_b$ is described by an attractor. The
post-blowup dynamics, $t > t_b$, is mapped to a different renormalized solution
starting infinitely far in the past. Consequently, it is associated with
another attractor. The $\nu$-regularization establishes a relation between
these two different "lives" of the renormalized system. We prove that, in some
generic situations, this procedure selects a unique global solution (or a
family of solutions), which does not depend on the details of the
regularization. We provide concrete examples and argue that these situations
are qualitatively similar to post-blowup scenarios observed in
infinite-dimensional models of turbulence.
