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arXiv · 2210.12125

Uniform upper bound for the number of limit cycles of planar piecewise linear differential systems with two zones separated by a straight line

Victoriano Carmona·Fernando Fernández-Sánchez·Douglas D. Novaes
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Abstract

The existence of a uniform upper bound for the maximum number of limit cycles of planar piecewise linear differential systems with two zones separated by a straight line has been subject of interest of hundreds of papers. After more than 30 years of investigation since Lum-Chua's work, it has remained an open question whether this uniform upper bound exists or not. Here, we give a positive answer for this question by establishing the existence of a natural number $L^*\leq 8$ for which any planar piecewise linear differential system with two zones separated by a straight line has no more than $L^*$ limit cycles. The proof is obtained by combining a newly developed integral characterization of Poincar\'{e} half-maps for linear differential systems with an extension of Khovanski\u{\i}'s theory for investigating the number of intersection points between smooth curves and a particular kind of orbits of vector fields.

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Victoriano Carmona, Fernando Fernández-Sánchez, Douglas D. Novaes. 2022-10-21. Uniform upper bound for the number of limit cycles of planar piecewise linear differential systems with two zones separated by a straight line. https://doi.org/10.1016/j.aml.2022.108501

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