Differential Equations With Codimension-N Discontinuity Sets

This paper investigates fundamental properties of a new class of dynamical systems, which are everywhere smooth except for a codimension-n discontinuity manifold with an arbitrary positive integer n. Such systems emerge naturally in modeling the motion of bodies with spatial point contacts as well as with finite contact surfaces under dry friction. As a special case, the investigated class includes Filippov systems (n = 1) as well as the recently introduced extended Filippov systems (n = 2). Trajectories reaching the discontinuity manifold are studied in detail, and new types of pathological behavior are uncovered, in systems where the local dynamics around the discontinuity manifold involves polycycles or strange attractors. The concept of crossing and sliding dynamics is extended for this type of system. The results are illustrated by several examples.