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

In this work we attempt to understand what behavior one should expect of a solution rajectory near Sigma when Sigma is attractive, what to expect when Sigma ceases to be attractive (at generic exit points), and finally we also contrast and compare the behavior of some regularizations proposed in the literature, whereby the piecewise smooth system is replaced by a smooth differential system. Through analysis and experiments in R-3 and R-4, we will confirm some known facts and provide some important insight: (i) when Sigma is attractive, a solution trajectory remains near Sigma, viz. sliding on Sigma is an appropriate idealization (though one cannot a priori decide which sliding vector field should be selected); (ii) when Sigma loses attractivity (at first order exit conditions), a typical solution trajectory leaves a neighborhood of (iii) there is no obvious way to regularize the system so that the regularized trajectory will remain near while 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 piecewise smooth system, without superimposing any assumption on what kind of dynamics near Sigma should have been taking place.