Limit Cycles of a Class of Discontinuous Planar Piecewise Linear Systems with Three Regions of Y-Type

  • Qianqian Zhao
  • Jiang YuEmail author


In this paper, we investigate the crossing limit cycles of a class of discontinuous planar piecewise linear dynamical systems with three regions of Y-type. We want to find the reason that cause the increase of the number of such limit cycles of the piecewise systems. We obtain the canonical form of such systems with focus–focus–focus type, and prove that the graph of Poincaré map of one of three subsystems has one inflection point. Naturally, we prove that the lower bound of the number of crossing limit cycles for this class systems is 3. We also give an example to simulate the four real crossing limit cycles.


Planar piecewise linear systems of Y-type Canonical form Poincaré maps Real crossing periodic orbits 

Mathematics Subject Classification

34C05 34C07 37G05 



The first author is partially supported by NNSF of China Grant Number 11501193. The second author is supported by NNSF of China Grant Numbers 11431008 and 11771282, NSF of Shanghai Grant Number 15ZR1423700.


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Mathematical SciencesShanghai Jiaotong UniversityShanghaiPeople’s Republic of China

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