A Filippov sliding vector field on an attracting co-dimension 2 discontinuity surface, and a limited loss-of-attractivity analysis

We consider sliding motion, in the sense of Filippov, on a discontinuity surface σ of co-dimension 2. We characterize, and restrict to, the case of σ being attractive through sliding. In this situation, we show that a certain Filippov sliding vector field fF (suggested in Alexander and Seidman, 1998 [2], di Bernardo et al., 2008 [6], Dieci and Lopez, 2011 [10]) exists and is unique. We also propose a characterization of first order exit conditions, clarify its relation to generic co-dimension 1 losses of attractivity for σ, and examine what happens to the dynamics on σ for the aforementioned vector field fF. Examples illustrate our results.