  In this paper we study zero-noise limits of $\alpha -$stable noise perturbed
ODE's which are driven by an irregular vector field $A$ with asymptotics $%
A(x)\sim \overline{a}(\frac{x}{\left\vert x\right\vert })\left\vert
x\right\vert ^{\beta -1}x$ at zero, where $\overline{a}>0$ is a continuous
function and $\beta \in (0,1)$. The results established in this article can be
considered a generalization of those in the seminal works of Bafico \cite% {Ba}
and Bafico, Baldi \cite{BB} to the multi-dimensional case.\ Our approach for
proving these results is inspired by techniques in \cite% {PP_self_similar} and
based on the analysis of an SDE for $t\longrightarrow \infty $, which is
obtained through a transformation of the perturbed ODE.
