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

Small Noise Perturbations in Multidimensional Case