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On Piecewise Linear Differential Systems with n Limit Cycles of Arbitrary Multiplicities in Two Zones

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Abstract

In this paper, we demonstrate rich dynamical phenomenon of piecewise linear differential systems having only two zones in the plane. We show that, for any given integer n and any integer tuple \(m=(m_1,m_2,\dots , m_n)\), \( m_i \ge 0 \), for \(i=1,\dots ,n\), there exists an aforementioned system which possesses exactly n limit cycles having multiplicities \(m_1\), \(m_2,\dots , m_n\), respectively. (i.e. there are totally \(m_1+m_2+\cdots +m_n\) limit cycles taking into account of multiplicities). Moreover, we can even choose the separation boundary of the zones to be an algebraic curve.

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Acknowledgements

We are particularly grateful for the many helpful comments and suggestions from the referee. This work is supported by NSCF of China 11671016, 11471027, 11371269 and Funds of Fujian Province Department of Education (JAT160082).

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Authors and Affiliations

  1. College of Mathematics and Computer Science, Fuzhou University, Fuzhou, 350108, China

    Changwu Zou

  2. School of Mathematics, Sun Yat-sen University, Zhuhai, 519082, Guangzhou, China

    Changjian Liu

  3. School of Mathematical Sciences, Peking University, Beijing, 100871, China

    Jiazhong Yang

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  1. Changwu Zou
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  2. Changjian Liu
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  3. Jiazhong Yang
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Correspondence to Jiazhong Yang.

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Zou, C., Liu, C. & Yang, J. On Piecewise Linear Differential Systems with n Limit Cycles of Arbitrary Multiplicities in Two Zones. Qual. Theory Dyn. Syst. 18, 139–151 (2019). https://doi.org/10.1007/s12346-018-0281-4

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  • DOI: https://doi.org/10.1007/s12346-018-0281-4

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