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We consider a piecewise smooth system in the neighborhood of a co-dimension 2discontinuity manifold $\Sigma$. Within the class of Filippov solutions, if$\Sigma$ is attractive, one should expect solution trajectories to slide on$\Sigma$. It is well known, however, that the classical Filippovconvexification methodology is ambiguous on $\Sigma$. The situation is furthercomplicated by the possibility that, regardless of how sliding on $\Sigma$ istaking place, during sliding motion a trajectory encounters so-called genericfirst order exit points, where $\Sigma$ ceases to be attractive.  In this work, we attempt to understand what behavior one should expect of asolution trajectory near $\Sigma$ when $\Sigma$ is attractive, what to expectwhen $\Sigma$ ceases to be attractive (at least, at generic exit points), andfinally we also contrast and compare the behavior of some regularizationsproposed in the literature.  Through analysis and experiments we will confirm some known facts, andprovide some important insight: (i) when $\Sigma$ is attractive, a solutiontrajectory indeed does remain near $\Sigma$, viz. sliding on $\Sigma$ is anappropriate idealization (of course, in general, one cannot predict whichsliding vector field should be selected); (ii) when $\Sigma$ loses attractivity(at first order exit conditions), a typical solution trajectory leaves aneighborhood of $\Sigma$; (iii) there is no obvious way to regularize thesystem so that the regularized trajectory will remain near $\Sigma$ as long as$\Sigma$ is attractive, and so that it will be leaving (a neighborhood of)$\Sigma$ when $\Sigma$ looses attractivity.  We reach the above conclusions by considering exclusively the given piecewisesmooth system, without superimposing any assumption on what kind of dynamicsnear $\Sigma$ (or sliding motion on $\Sigma$) should have been taking place.

Piecewise smooth systems near a co-dimension 2 discontinuity manifold: can one say what should happen?