Second-order Filippov systems don't have sliding regions.
In standard discontinuous dynamical systems, when a trajectory hits a boundary between two regimes, it can get stuck — sliding along the boundary because both vector fields push toward it. This “sliding mode” is fundamental to Filippov's classical theory. Control engineers exploit it; theorists analyze it; the phenomenon is central to how discontinuous systems are understood.
Galvanetto, Magri, and Piiroinen (arXiv:2603.10333) develop the theory for second-order Filippov systems — where the equation is second-order, like mechanical oscillators — and find that the discontinuity surface consists only of crossing regions. No sliding regions exist. Trajectories always cross the boundary.
The dynamics live instead on “tangency surfaces” — lower-dimensional sets where both vector fields are tangent to the boundary simultaneously. These aren't sliding regions in the classical sense. They're boundaries of the crossing regions, and orbits can be attracted to or repelled from them, but the mechanism is fundamentally different.
The structural reason: a second-order equation imposes continuity of the state (position) even when the derivative (velocity) is discontinuous. This extra smoothness constraint eliminates the geometric configuration that creates sliding. The boundary can't trap trajectories because the second-order structure forces them through.
This is a case where a constraint that seems to reduce the system's behavior — second-order structure limits what can happen at boundaries — actually produces richer dynamics at a different location. The sliding moves from the obvious place (the boundary surface) to the subtle place (the tangency curves). The phenomenon doesn't disappear; it relocates to where the constraint permits it. Adding structure doesn't simplify; it rearranges where the complexity lives.
Galvanetto, Magri, and Piiroinen, "Second-order Filippov systems: sliding dynamics without sliding regions," arXiv:2603.10333 (2026).