This talk considers ordinary differential equations with discontinuous right-hand side, where the discontinuity of the vector field takes place on smooth hyper-surfaces of the phase space. The main emphasis is put on what happens in the intersection of two of these surfaces. Solutions may traverse a hyper-surface, but they can also stick in them (sliding mode). There the concept of solution is extended by considering convex combinations of the adjacent vector fields (Filippov's approach).
When a solution enters the intersection of two hyper-surfaces, its continuation is often not unique. A natural selection criterium is to consider only solutions that can be considered as the limit, for $\varepsilon \to 0$, of the solution of a differential equation (regularization), where the vector field is smoothed out in an $\varepsilon$-neighbourhood of the discontinuity manifolds.
A complete classification of possible transitions of solutions close to the intersection of two discontinuity surfaces is given. The result is sometimes counterintuitive. Moreover, the new insight permits, in the presence of high oscillations in solutions of the regularized differential equation, to propose a simple modification that suppresses these high oscillations and makes the numerical treatment much more efficient.
This is joint work with Nicola Guglielmi from the University of L'Aquila.