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On the Number of Limit Cycles in General Planar Piecewise Linear Differential Systems with Two Zones Having Two Real Equilibria

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Abstract

A general family of planar piecewise linear ODEs with two zones both having a real focus and separated by a straight line is considered. By analyzing the number of zero points of a new function related to the intersection points of the trajectories of the linear subsystems with the separation line, complete results on the existence and number of limit cycles are obtained. In particular, complete parameter regions for the existence of 1–2 limit cycles are provided with two concrete examples, which will be helpful in studying some kinds of discontinuity-induced bifurcations (i.e., DIBs). Based on the main results, it is obtained that the family of planar piecewise linear ODEs with focus–focus dynamics separated by a straight line can have 3 limit cycles if and only if one subsystem has a real focus and the other one has a virtual focus. Moreover, it is showed that five is the least value of the number of parameters needed in canonical forms of such systems.

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Acknowledgements

This work is supported by National Natural Science Foundation of China (11301196) and the Fundamental Research Funds for the Central Universities, HUST (2015QN128).

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  1. Hubei Key Laboratory of Engineering Modeling and Scientific Computing, School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, Hubei, People’s Republic of China

    Song-Mei Huan

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Correspondence to Song-Mei Huan.

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Huan, SM. On the Number of Limit Cycles in General Planar Piecewise Linear Differential Systems with Two Zones Having Two Real Equilibria. Qual. Theory Dyn. Syst. 20, 4 (2021). https://doi.org/10.1007/s12346-020-00441-x

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