Regularization and Geometry of Piecewise Smooth Systems with Intersecting Discontinuity Sets

In this work, we study the dynamics of piecewise smooth (PWS) systems on a codimension-2 transverse intersection of two codimension-1 discontinuity sets. The Filippov convention can be extended to such intersections, but this approach does not provide a unique sliding vector and, as opposed to the classical sliding vector field on codimension-1 discontinuity manifolds, there is no agreed notion of stability in the codimension-2 context.

From a modeling perspective, one may interpret this lack of determinacy as a fact that additional modeling is required; knowing the four adjacent vector-fields is not enough to define a unique forward flow. In this paper, we provide additional information to the system by performing a regularization of the PWS system, introducing two regularization functions and a small perturbation parameter.

Then, based on singular perturbation theory, we define sliding and stability of sliding through a critical manifold of the singularly perturbed, regularized system. We show that this notion of sliding vector field coincides with the Filippov one. The regularized system gives a parameterized surface, the canopy [M.

R. Jeffrey, SIAM J. Appl. Dyn. Syst., 13 (2014), pp. 1082--1105], independent of the regularization functions. This surface serves as our natural basis to derive new and simple geometric criteria on the existence, multiplicity, and stability of the sliding flow, depending only on the smooth vector-fields around the intersection.

Interestingly, we are able to show that if there exist two sliding vector-fields, then one is a saddle and the other is of focus/node/center type. This means that there is at most one stable sliding vector field. We then investigate the effect of the choice of the regularization functions, and, using a blowup approach, we demonstrate the mechanisms through which sliding behavior can appear or disappear on the intersection and describe what consequences this has on the dynamics on the adjacent codimension-1 discontinuity sets.

This blowup method also shows that the PWS limit of the regularization may be well-defined, even in cases where the Filippov sliding vector field is nonunique. Finally, we show the existence of canard explosions of regularizations of PWS systems in $\mathbb{R}^3$ that depend on a single unfolding parameter.