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A Complete Classification on the Center-Focus Problem of a Generalized Cubic Kukles System with a Nilpotent Singular Point

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Abstract

In this paper, we study the center-focus problem for a generalized cubic Kukles system with a nilpotent singular point, which consists of a cubic system with an extra 4th-order term. A complete classification is given on the center conditions which are explicitly expressed in term of the system parameters. A total of 15 cases are obtained, among them 4 for the generalized cubic Kukles system and 12 for the cubic Kukles system, with one common for both. One of the center conditions is analytic. Moreover, it is shown that 8 small-amplitude limit cycles can bifurcate in the neighborhood of the singular point for the generalized cubic Kukles system, while only 7 small-amplitude limit cycles can exist around the singular point for the cubic Kukles system. The center-focus problem for the generalized cubic Kukles system with a nilpotent origin is thoroughly solved.

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Acknowledgements

This research was partially supported by National Natural Science Foundation of China, No. 12071198 (F. Li), and the Natural Sciences and Engineering Research Council of Canada, No. R2686A02 (P. Yu).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Linyi University, Linyi, 276005, Shandong, China

    Feng Li & Yuanyuan Liu

  2. School of Statistics and Mathematics, Guangdong University of Finance and Economics, Guangzhou, 510320, Guangdong, China

    Ting Chen

  3. Department of Mathematics, Western University, London, ON, N6A 5B7, Canada

    Pei Yu

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  1. Feng Li
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  2. Ting Chen
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Contributions

Feng Li: Conceptualization, methodology, computation. Ting Chen: Program verification. Yuanyuan Liu: Check some calculations and proofreading. Pei Yu: Supervision, computation, writing, reviewing and editing.

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Correspondence to Pei Yu.

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Li, F., Chen, T., Liu, Y. et al. A Complete Classification on the Center-Focus Problem of a Generalized Cubic Kukles System with a Nilpotent Singular Point. Qual. Theory Dyn. Syst. 23, 8 (2024). https://doi.org/10.1007/s12346-023-00863-3

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