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A Complement to the Uniqueness of the Limit Cycle of a Family of Systems with Homogeneous Components

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Abstract

Consider the number of limit cycles of a family of systems with homogeneous components: \( {\dot{x}}=y, {\dot{y}}=-x^3+\alpha x^2y+y^3. \) We show that there is an \(\alpha ^*<0\) such that the system has exactly one limit cycle for \(\alpha \in (\alpha ^*,0),\) while no limit cycle for the else region. This completes a previous result and also gives a positive answer to the second part of Gasull’s 3rd problem listed in the paper (SeMA J 78(3):233–269, 2021). To obtain this result, we mainly analyse the behavior of the heteroclinic separatrices at infinity.

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (No. 12171491). The authors would like to thank the editor Jaume Giné for handling the submission and the anonymous reviewers for their useful comments.

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ZZ wrote the main manuscript text. All authors reviewed the manuscript.

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Correspondence to Ziwei Zhuang.

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Zhuang, Z., Liu, C. A Complement to the Uniqueness of the Limit Cycle of a Family of Systems with Homogeneous Components. Qual. Theory Dyn. Syst. 23, 205 (2024). https://doi.org/10.1007/s12346-024-00991-4

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