Applicability Conditions for TDM, ZDM, and PDM in Filippov Systems

Determine the precise conditions under which the transverse discontinuity map (TDM), zero-time discontinuity mapping (ZDM), or Poincaré discontinuity mapping (PDM) should be employed to analyze trajectory interactions with switching manifolds in Filippov piecewise-smooth dynamical systems, including criteria that distinguish their validity for transverse crossings versus near-grazing impacts.

Background

Filippov systems are piecewise-smooth dynamical systems whose trajectories can interact with switching manifolds either transversely or near-grazing, giving rise to discontinuity-induced bifurcations. Analytical tools such as the transverse discontinuity map (TDM), zero-time discontinuity mapping (ZDM), and Poincaré discontinuity mapping (PDM) are commonly used to relate pre- and post-impact states.

The paper shows that the commonly used first-order (linearized) saltation matrix embedded in standard TDM approaches can produce inaccurate predictions near grazing, motivating a higher-order TDM that better captures flight times and impact conditions. Despite these advances, there is no established guideline delineating when TDM, ZDM, or PDM should be preferred in Filippov systems, particularly across regimes of transverse versus near-grazing interactions.

References

Consequently, the appropriate conditions under which the TDM, ZDM, or PDM should be employed in Filippov systems remain an open question.

— Higher-Order Transverse Discontinuity Mapping in Filippov Systems: Analysis and Experimental Validation using an Electronic Circuit  (2506.18379 - Chawla et al., 23 Jun 2025) in Introduction, paragraph following Eq. (1)

For a point $x_0 \in \Omega_0$ satisfying $L_f h(x_0) L_g h(x_0)=0$, it remains an open problem whether there exists a trajectory of $F$ starting from $x_0.

— Differential inclusions and the chattering phenomenon  (2610.03561 - Ivanov, 2 Oct 2026) in Section 1, subsection “Previous results”