The complex center and complex isochronous center problems for a complex quartic polynomial system

Zhanghan Huang; Jingping Lu; Zhanghan Huang; Jingping Lu

doi:10.3934/nhm.2026010

Networks and Heterogeneous Media

2026, Volume 21, Issue 1: 213-233. doi: 10.3934/nhm.2026010

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The complex center and complex isochronous center problems for a complex quartic polynomial system

Zhanghan Huang ¹,
Jingping Lu ^{2,3
,
,}

1.
School of Media and Art Design, Guilin University of Aerospace Technology, Guilin 541004, China
2.
School of Science, Guilin University of Aerospace Technology, Guilin 541004, China
3.
School of Mathematics and Statistics, Guilin University of Technology, Guilin 541004, China

Received: 09 September 2025 Revised: 22 January 2026 Accepted: 30 January 2026 Published: 05 February 2026

In this paper, we first study the complex center and complex isochronous center problems for a complex quartic polynomial differential system. More precisely, by calculating and decomposing the variety of the ideal generated by the singular point quantities (resp. complex period quantities), we obtain the necessary conditions for the resonant elementary equilibrium (resp. complex center) of the system to be a complex center (resp. a complex isochronous center). Using time-reversibility and the Darboux integrable theory, we rigorously prove that these conditions are also sufficient. Furthermore, when the coefficients and the variables of the system are complex conjugates, we not only derive the necessary and sufficient conditions for the center-type equilibrium to be both a center and an isochronous center, but we also give the parametric conditions under which the center-type equilibrium of the system becomes a weak focus of order 10. In this case, although the independence of the focal quantities is not satisfied, we still prove that there exist parametric conditions under which 10 limit cycles can bifurcate from this weak focus.
- quartic polynomial differential system,
- center,
- isochronous center,
- limit cycle,
- focal quantities
Citation: Zhanghan Huang, Jingping Lu. The complex center and complex isochronous center problems for a complex quartic polynomial system[J]. Networks and Heterogeneous Media, 2026, 21(1): 213-233. doi: 10.3934/nhm.2026010

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Abstract

In this paper, we first study the complex center and complex isochronous center problems for a complex quartic polynomial differential system. More precisely, by calculating and decomposing the variety of the ideal generated by the singular point quantities (resp. complex period quantities), we obtain the necessary conditions for the resonant elementary equilibrium (resp. complex center) of the system to be a complex center (resp. a complex isochronous center). Using time-reversibility and the Darboux integrable theory, we rigorously prove that these conditions are also sufficient. Furthermore, when the coefficients and the variables of the system are complex conjugates, we not only derive the necessary and sufficient conditions for the center-type equilibrium to be both a center and an isochronous center, but we also give the parametric conditions under which the center-type equilibrium of the system becomes a weak focus of order 10. In this case, although the independence of the focal quantities is not satisfied, we still prove that there exist parametric conditions under which 10 limit cycles can bifurcate from this weak focus.

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