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Some properties of Melnikov functions near a cuspidal loop

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Abstract

In this paper, we consider the first-order Melnikov functions and limit cycle bifurcations of a near-Hamiltonian system near a cuspidal loop. By establishing relations between the coefficients in the expansions of the two Melnikov functions, we give a general method to obtain the number of limit cycles near the cuspidal loop. As an application, we consider a kind of Liénard systems and obtain a new estimation on the lower bound of the maximum number of limit cycles.

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Acknowledgements

The first author was supported by National Natural Science Foundation of China (Grant No. 11971145). The second author was supported by National Natural Science Foundation of China (Grant No. 11931016) and the National Key R&D Program of China (Grant No. 2022YFA1005900). The authors thank the referees for their helpful suggestions, which have greatly helped improve the presentation of this paper.

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Authors and Affiliations

  1. School of Mathematical Sciences, Hebei Normal University, Shijiazhuang, 050024, China

    Junmin Yang

  2. Department of Mathematics, Zhejiang Normal University, **hua, 321004, China

    Maoan Han

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  1. Junmin Yang
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  2. Maoan Han
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Correspondence to Maoan Han.

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Yang, J., Han, M. Some properties of Melnikov functions near a cuspidal loop. Sci. China Math. 67, 767–786 (2024). https://doi.org/10.1007/s11425-022-2124-7

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  • DOI: https://doi.org/10.1007/s11425-022-2124-7

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