不连续平面分段线性系统的两点和四点极限环

Abstract

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

CLC Number:

O175

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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Two-Point and Four-Point Limit Cycles in Discontinuous Planar Piecewise Linear Systems

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