The stability of boundary equilibria of three-dimensional Filippov systems

For three-dimensional piecewise-smooth systems of ordinary differential equations, this paper characterises the stability of points that belong to a switching surface and are equilibria of exactly one of the two neighbouring pieces of the system. Stability is challenging to characterise when nearby orbits repeatedly switch between regular motion on one side of the switching surface, and sliding motion on the switching surface, as defined via Filippov's convention. We prove that in this case stability is governed by the behaviour of a global reinjection mechanism of a four-parameter family of piecewise-linear hybrid systems, and perform a detailed numerical study of this family.