Uniform limit-cycle bound for Sotomayor–Teixeira regularizations of piecewise-linear systems

Determine whether there exists a uniform upper bound on the number of limit cycles in Sotomayor–Teixeira regularizations of piecewise-linear planar systems.

Background

The paper places this question in the context of the second part of Hilbert’s sixteenth problem, which asks for uniform bounds on the number of limit cycles in specified classes of planar systems. Although regularizations of piecewise quadratic systems can produce arbitrarily many hyperbolic limit cycles, the corresponding question for regularizations of piecewise-linear systems remains unresolved.

The present work addresses only sliding cycles associated with generic visible-visible and visible-invisible two-fold configurations, under the assumption that the slow divergence integral does not vanish identically. Thus, the broader uniform-bound question is not settled in full by the paper.

References

It remains an open problem whether there exists a uniform upper bound on the number of limit cycles in Sotomayor--Teixeira regularizations of PWL systems, see .

— Cyclicity of sliding cycles in regularizations of piecewise linear two-folds  (2609.18561 - Huzak et al., 16 Sep 2026) in Section 1, Introduction and statement of the problem

The periodic orbits of system $Z_{\varepsilon,\varepsilon \widetilde{\lambda}{\phi}$ in the case $I{a,b} \equiv 0$ are not considered in the present paper. We leave this degenerate case for future work, since its analysis is expected to depend on the choice of regularization function $\phi$.

— Cyclicity of sliding cycles in regularizations of piecewise linear two-folds  (2609.18561 - Huzak et al., 16 Sep 2026) in Section 1, Introduction and statement of the problem; immediately after Theorem mhtm-c in Section 2