Algebraic limit cycles of planar discontinuous piecewise linear differential systems with an angular switching boundary

被引：0
作者：
Llibre, Jaume [1 ]
Xiong, Haichao [2 ]
Zhang, Weinian [2 ]
机构：
[1] Univ Autonoma Barcelona, Dept Matemat, Barcelona 08193, Catalonia, Spain
[2] Sichuan Univ, Dept Math, Chengdu 610064, Sichuan, Peoples R China
来源：
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS | 2025年 / 546卷 / 02期
基金：
国家重点研发计划; 中国国家自然科学基金;
关键词：
Algebraic limit cycle; Polynomial differential systems; Polynomial first integral; Rational first integral; 16TH HILBERT PROBLEM; MATHEMATICAL PROBLEMS; BIFURCATIONS; EQUATIONS; NUMBER;
D O I：
10.1016/j.jmaa.2025.129224
中图分类号：
O29 [应用数学];
学科分类号：
070104 ;
摘要：
Known results show that, with a 9-angular switching boundary for 9 is an element of (0, pi], a planar piecewise linear differential system formed by two Hamiltonian linear subsystems has no crossing algebraic limit cycles of type I, i.e., those cycles crossing one of the two sides of the 9-angular switching boundary twice only, and at most two crossing algebraic limit cycles of type II, i.e., those cycles crossing both sides of the 9-angular switching boundary once separately. In this paper, using the Chebyshev theory and Descartes' rule to overcome difficulties in applying Gr & ouml;bner basis to solve polynomial systems, we study the number of crossing algebraic limit cycles for such a piecewise linear system having a Hamiltonian sub-system and a non-Hamiltonian sub-system. We prove that the maximum number of type I is one and the lower and upper bounds of the maximum number of type II are five and seven, respectively, and show the coexistence of type I and type II, which implies that a lower bound for the maximum number of all crossing algebraic limit cycles is six. (c) 2025 Published by Elsevier Inc.
引用
收藏
页数：31
相关论文
共 45 条
  • [1] Berezin I.S., Zhidkov N.P., Computing Methods, (1965)
  • [2] di Bernardo M., Budd C.J., Champneys A.R., Kowalczyk P., Piecewise-Smooth Dynamical Systems, Theory and Applications, (2008)
  • [3] Buzzi C., Gasull A., Torregrosa J., Algebraic limit cycles in piecewise linear differential systems, Int. J. Bifurc. Chaos, 28, (2018)
  • [4] Buzzi C., Pessoa C., Torregrosa J., Piecewise linear perturbations of a linear center, Discrete Contin. Dyn. Syst., 33, pp. 3915-3936, (2013)
  • [5] Cardin P.T., Torregrosa J., Limit cycles in planar piecewise linear differential systems with nonregular separation line, Physica D, 337, pp. 67-82, (2016)
  • [6] Carmona V., Fernandez-Sanchez F., Novaes D.D., Uniform upper bound for the number of limit cycles of planar piecewise linear differential systems with two zones separated by a straight line, Appl. Math. Lett., 137, (2023)
  • [7] Chen X., Llibre J., Zhang W., Averaging approach to cyclicity of Hopf bifurcation in planar linear-quadratic polynomial discontinuous differential systems, Discrete Contin. Dyn. Syst., Ser. B, 22, pp. 3953-3965, (2017)
  • [8] Christopher C., Polynomial vector fields with prescribed algebraic limit cycles, Geom. Dedic., 88, pp. 255-258, (2001)
  • [9] Cox D., Little J., O'Shea D., Ideals, Varieties, and Algorithms, (1997)
  • [10] da Cruz L.P.C., Torregrosa J., Simultaneous bifurcation of limit cycles from a cubic piecewise center with two period annuli, J. Math. Anal. Appl., 461, pp. 248-272, (2018)
12345