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In this work, we study the dynamics of piecewise smooth systems on acodimension-2 transverse intersection of two codimension-1 discontinuity sets.The Filippov convention can be extended to such intersections, but thisapproach does not provide a unique sliding vector and, as opposed to theclassical sliding vector-field on codimension-1 discontinuity manifolds, thereis no agreed notion of stability in the codimension-2 context. In this paper,we perform a regularization of the piecewise smooth system, introducing tworegularization functions and a small perturbation parameter. Then, based onsingular perturbation theory, we define sliding and stability of slidingthrough a critical manifold of the singularly perturbed, regularized system. Weshow that this notion of sliding vector-field coincides with the Filippov one.The regularized system gives a parameterized surface (the canopy) independentof the regularization functions. This surface serves as our natural basis toderive new and simple geometric criteria on the existence, multiplicity andstability of the sliding flow, depending only on the smooth vector fieldsaround the intersection. Interestingly, we are able to show that if there existtwo sliding vector-fields then one is a saddle and the other is offocus/node/center type. This means that there is at most one stable slidingvector-field. We then investigate the effect of the choice of theregularization functions, and, using a blowup approach, we demonstrate themechanisms through which sliding behavior can appear or disappear on theintersection and describe what consequences this has on the dynamics on theadjacent codimension-1 discontinuity sets. Finally, we show the existence ofcanard explosions of regularizations of PWS systems in $\mathbb R^3$ thatdepend on a single unfolding parameter.

Regularization and geometry of piecewise smooth systems with intersecting discontinuity sets