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One of the most common hypotheses on the theory of non-smooth dynamicalsystems is a regular surface as switching manifold, at which case there is atleast well-defined and established Filippov dynamics. However, systems withsingular switching manifolds still lack such well-established dynamics,although present in many relevant models of phenomena where multiple switchesor multiple abrupt changes occur. At this work, we leverage a methodology that,through blow-ups and singular perturbation, allows the extension of Filippovdynamics to the singular case. Specifically, tridimensional systems whoseswitching manifold consists of an algebraic manifold with transversalself-intersection are considered. This configuration, known as doublediscontinuity, represents systems with two switches and whose singular partconsists of a straight line, where ordinary Filippov dynamics is not directlyapplicable. For the general, non-linear case, beyond defining the so-calledfundamental dynamics over the singular part, general theorems on itsqualitative behavior are provided. For the affine case, however, theorems fullydescribing the fundamental dynamics are obtained. Finally, this fine-grainedcontrol over the dynamics is leveraged to derive Peixoto like theoremscharacterizing semi-local structural stability.

Dynamics and Stability of Non-Smooth Dynamical Systems with Two Switches