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Existence of Four-Crossing-Points Limit Cycles in Planar Sector-Wise Linear Systems with Saddle-Saddle Dynamics

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Abstract

In this paper, we study a family of planar piecewise linear systems with saddle-saddle dynamics and sector-wise separation, which is formed by two rays starting from the same point. We mainly investigate the existence of four-crossing-points limit cycles, which intersect each of the two separation rays at two points. This is a new and complex type of limit cycle that cannot exist in planar piecewise linear systems with two zones separated by a straight line and their existence in planar sector-wise linear systems with saddle-saddle dynamics has not been studied until now. We first obtain some sufficient and necessary conditions for the existence of a special four-crossing-points limit cycle. Then based on this result, we give some sufficient conditions for the existence of general four-crossing-points limit cycles. Moreover, we show that the four-crossing-points limit cycle and two different types of two-crossing-points limit cycles can exist simultaneously by providing concrete examples.

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Data Availability

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

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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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Authors and Affiliations

  1. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, Hubei, People’s Republic of China

    Xiao-Juan Liu, Xiao-Song Yang & Song-Mei Huan

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

    Song-Mei Huan

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  1. Xiao-Juan Liu
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We certify that this manuscript is original and has not been published and will not be submitted elsewhere for publication while being considered by Qualitative Theory of Dynamical Systems. And the study is not split up into several parts to increase the quantity of submissions and submitted to various journals or to one journal over time. No data have been fabricated or manipulated (including images) to support our conclusions. No data, text, or theories by others are presented as if they were our own. This article does not contain any studies with human participants or animals.

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Liu, XJ., Yang, XS. & Huan, SM. Existence of Four-Crossing-Points Limit Cycles in Planar Sector-Wise Linear Systems with Saddle-Saddle Dynamics. Qual. Theory Dyn. Syst. 21, 63 (2022). https://doi.org/10.1007/s12346-022-00582-1

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