  One of the most common hypotheses on the theory of non-smooth dynamical
systems is a regular surface as switching manifold, at which case there is at
least well-defined and established Filippov dynamics. However, systems with
singular switching manifolds still lack such well-established dynamics,
although present in many relevant models of phenomena where multiple switches
or multiple abrupt changes occur. At this work, we leverage a methodology that,
through blow-ups and singular perturbation, allows the extension of Filippov
dynamics to the singular case. Specifically, tridimensional systems whose
switching manifold consists of an algebraic manifold with transversal
self-intersection are considered. This configuration, known as double
discontinuity, represents systems with two switches and whose singular part
consists of a straight line, where ordinary Filippov dynamics is not directly
applicable. For the general, non-linear case, beyond defining the so-called
fundamental dynamics over the singular part, general theorems on its
qualitative behavior are provided. For the affine case, however, theorems fully
describing the fundamental dynamics are obtained. Finally, this fine-grained
control over the dynamics is leveraged to derive Peixoto like theorems
characterizing semi-local structural stability.
