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

被引：20

作者：

Carmona, Victoriano ^{[1
,2
]}

Fernandez-Sanchez, Fernando ^{[3
,4
]}

Novaes, Douglas D. ^{[5
]}

机构：

[1] Univ Seville, Escuela Politecn Super, Dept Matemat Aplicada 2, Calle Virgen Africa 7, Seville 41011, Spain

[2] Univ Seville, Escuela Politecn Super, IMUS, Calle Virgen Africa 7, Seville 41011, Spain

[3] Univ Seville, Escuela Tecn Super Ingn, Dept Matemat Aplicada 2, Camino Descubrimientos S-N, Seville 41092, Spain

[4] Univ Seville, Escuela Tecn Super Ingn, IMUS, Camino Descubrimientos S-N, Seville 41092, Spain

[5] Univ Estadual Campinas UNICAMP, Inst Matemat Estat & Comp Cient IMECC, Dept Matemat, Rua Sergio Buarque de Holanda 651, BR-13083859 Campinas, SP, Brazil

来源：

APPLIED MATHEMATICS LETTERS | 2023年 / 137卷

基金：

巴西圣保罗研究基金会;

关键词：

Planar piecewise differential linear; systems; Limit cycles; Upper bounds; Poincar? half-maps; Khovanski???s theory;

D O I：

10.1016/j.aml.2022.108501

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

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* <= 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 Poincare half-maps for linear differential systems with an extension of Khovanskii's theory for investigating the number of intersection points between smooth curves and a particular kind of orbits of vector fields. (c) 2022 Elsevier Ltd. All rights reserved.

引用

页数：8

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