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Limit Cycles in the Discontinuous Planar Piecewise Linear Systems with Three Zones

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Abstract

In this paper, we investigate the existence of limit cycles for the discontinuous planar piecewise linear systems with three zones separated by two parallel straight lines. Based on the methods of first integral and Poincaré map, we present the maximum number of limit cycles in the normal forms of systems with boundary focus-center-boundary focus and boundary focus-center-center types, respectively. Then we show that such discontinuous piecewise linear systems can have at most three limit cycles, being two of them of four intersection points type and the third one of two intersection points type.

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Acknowledgements

The authors would like to express their appreciation to the reviewers for their helpful comments and suggestions. All authors also would like to thank Professor Xiang Zhang for many valuable suggestions leading to the completion of this paper.

Funding

This work was partially funded by the National Natural Science Foundation of China under Grant No. 11871022 and Shanghai Key Laboratory of PMMP, and supported in part by Science and Technology Commission of Shanghai Municipality (No. 18dz2271000).

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  1. Department of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai, 200241, People’s Republic of China

    Zhengkang Li & Xingbo Liu

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  1. Zhengkang Li
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Correspondence to Xingbo Liu.

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Li, Z., Liu, X. Limit Cycles in the Discontinuous Planar Piecewise Linear Systems with Three Zones. Qual. Theory Dyn. Syst. 20, 79 (2021). https://doi.org/10.1007/s12346-021-00496-4

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