Classification of Hidden Dynamics in Discontinuous Dynamical Systems

Ordinary differential equations with discontinuous right-hand side, where the discontinuity of the vector field arises on smooth surfaces of the phase space, are the topic of this work. The main emphasis is the study of solutions close to the intersection of two discontinuity surfaces. There, the so-called hidden dynamics describe the smooth transition from ingoing to outgoing solution directions, which occurs instantaneously in the jump discontinuity of the vector field. This paper presents a complete classification of such transitions (assuming the vector fields surrounding the intersection are transversal to it). Since the hidden dynamics are realized by standard space regularizations, much insight is obtained for them. One can predict, in the case of multiple solutions of the discontinuous problem, which solution (classical or sliding mode) will be approximated after entering the intersection of two discontinuity surfaces. A novel modification of space regularizations is presented that permits us to avoid (unphysical) high oscillations and makes a numerical treatment more efficient.