Classification of hidden dynamics in discontinuous dynamical systems
N. Guglielmi and E. Hairer
Abstract. 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 describes the smooth transition
from ingoing to outgoing
solution directions, which occurs instantaneously in the jump discontinuity
of the vector field. This article presents a complete classification
of such transitions.
Since the hidden dynamics is 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 to avoid (unphysical) high oscillations and makes a numerical treatment more
efficient.
Key Words. Discontinuous vector fields, regularization, asymptotic expansions, hidden dynamics, stabilization.