Sliding modes of high codimension in piecewise-smooth dynamical systems
Nicola Guglielmi and Ernst Hairer
Abstract. We consider piecewise-smooth dynamical systems, i.e.,
systems of ordinary differential equations switching between different
sets of equations on distinct domains, separated by hyper-surfaces.
As is well-known, when the solution approaches a discontinuity manifold,
a classical solution may cease to exist.
For this reason, starting with the pioneering work of Filippov, a concept of
weak solution (also known as sliding mode) has been introduced and studied.
Nowadays, the solution of piecewise-smooth dynamical systems in and close to
discontinuity manifolds is well understood, if the manifold consists locally
of a single discontinuity hyper-surface or of the intersection of two
discontinuity hyper-surfaces. The present work presents partial results
on the solution in and close to discontinuity manifolds of codimension 3 and higher.
Key Words. piecewise-smooth systems, Filippov solution, sliding modes
in high codimension, regularization, hidden dynamics