Discontinuity mappings for vector fields with higher order continuity

摘要

This paper concerns the derivation of discontinuity mappings in dynamical systems defined by piecewise smooth vector fields. The cases of trajectories that either cross or are quadratically tangent to a co-dimension 1 surface in state space are treated. The vector field is assumed to have C~(n―1) continuity across the surface, and it is shown that the discontinuity mappings are equal to the identity up to order n, and the structure of the mappings and the lowest order term is derived. An example where the mapping can be derived in another way is used to illustrate the theory.