不连续平面分段线性系统的两点和四点极限环

摘要/Abstract

摘要：

该文研究一类具有折线边界的不连续平面分段线性系统中两点和四点极限环的存在性、共存性及最大共存个数. 文献[29, 30] (Llibre & Teixeira, 2017 & 2018) 提出了两个公开问题: 无平衡点或仅具有中心型平衡点的平面分段线性系统是否存在极限环? 该文假设两个子系统由无平衡点的线性 Hamiltonian 系统或具有中心型平衡点的线性系统构成, 利用首次积分方法, 证明了与折线边界交于两个点的两点极限环的最大个数为 2, 与折线边界交于四个点的四点极限环的最大个数为 1. 在 1 个四点极限环存在的前提下, 仅具有唯一的两点极限环可以与其共存. 此外, 该文还利用数值模拟提供了精确的数值结果.

关键词: 极限环, 不连续平面分段线性系统, 首次积分, Hamiltonian 系统, 中心型平衡点

Abstract:

In this paper, we study the existence, coexistence and maximum number of coexisting elements for two-point and four-point limit cycles in discontinuous planar piecewise linear systems separated by nonregular separation line. Refs. [29, 30] (Llibre & Teixeira, 2017 & 2018) posed two open problems: Can piecewise linear differential systems without equilibria or with only centers produce limit cycles? Assume that two subsystems are composed of a Hamiltonian system without equilibrium points or a linear system with center type equilibrium. Via the method of first integral, it is proved that the maximum number of two-point limit cycles that intersect with nonregular separation line boundary at two points is 2, and the maximum number of four point limit cycles that intersect with nonregular separation line boundary at four points is 1. Under the premise of the existence of one four-point limit cycle, only a unique two-point limit cycle could coexist with it. In addition, we also provides accurate numerical results by numerical simulations.

Key words: limit cycle, discontinuous planar piecewise linear system, first integral, Hamiltonian system, center type equilibrium

中图分类号:

O175

李争康. 不连续平面分段线性系统的两点和四点极限环[J]. 数学物理学报, 2025, 45(3): 824-842.

Li Zhengkang. Two-Point and Four-Point Limit Cycles in Discontinuous Planar Piecewise Linear Systems[J]. Acta mathematica scientia,Series A, 2025, 45(3): 824-842.

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