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<p>In this paper, we investigate the limit cycles of a class of switching ordinary differential equations consisting of two sub-equations. We propose a general framework for studying the maximum number of limit cycles of the equations. As applications, we analyze three biological models in recent literature. We prove that a general model of single species with seasonal constant-yield harvesting can only possess at most two limit cycles, which improves the work of Xiao (<CitationRef CitationID="CR41">2016</CitationRef>) and Han et&#xa0;al. (<CitationRef CitationID="CR19">2018</CitationRef>). We also apply our framework to a general model described by the Abel equations with periodic step function coefficients, showing that its maximum number of limit cycles is three. Finally, a population suppression model for mosquitoes considered by Yu (<CitationRef CitationID="CR44">2020</CitationRef>) and Zheng et&#xa0;al. (<CitationRef CitationID="CR53">2021</CitationRef>) is studied using our approach.</p>

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On the Limit Cycles of Switching Ordinary Differential Equations

  • 期刊论文

  • 发表日期：

    2026年5月29日

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  • Renhao Tian,
  • Jianfeng Huang,
  • Yulin Zhao

摘要

In this paper, we investigate the limit cycles of a class of switching ordinary differential equations consisting of two sub-equations. We propose a general framework for studying the maximum number of limit cycles of the equations. As applications, we analyze three biological models in recent literature. We prove that a general model of single species with seasonal constant-yield harvesting can only possess at most two limit cycles, which improves the work of Xiao (2016) and Han et al. (2018). We also apply our framework to a general model described by the Abel equations with periodic step function coefficients, showing that its maximum number of limit cycles is three. Finally, a population suppression model for mosquitoes considered by Yu (2020) and Zheng et al. (2021) is studied using our approach.