Boundary intersection crossing bifurcation in the presence of sliding

Piecewise smooth systems are known to present a richer set of bifurcations than their smooth counterparts. An interesting family of bifurcations that is present in this type of systems are the so called boundary intersection crossing bifurcations, that take place when a periodic orbit crosses the intersection between two or more discontinuity boundaries. Such bifurcations have been observed in many different models, and have been studied in a number of papers over the past few years. Nonetheless, the particular case in which sliding solutions (as defined by Filippov) are involved, has been left out in previous analyses. This paper addresses this particular case, carrying out a complete analysis and deriving the discontinuity mappings that can be used to characterise such bifurcations. Then, in the second part of the paper, the results are applied to the study of a model of a common electronic device, showing how the mappings can be used systematically to determine the dynamics around the bifurcation.