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Complete solution to the weak persistent center and isochronous center problems of cubic polynomial systems

  • Feng Li
  • Huimei Liu
  • Chen Cao
  • Pei Yu
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Abstract

In this paper, we investigate weakly persistent centers for several classes of planar cubic differential systems, including systems with a nondegenerate linear center and systems with a nilpotent center. Necessary parameter relations are obtained by computing singular point quantities or quasi-Lyapunov constants, while their sufficiency is established by suitable analytic constructions. For the general complex cubic system, we obtain necessary and sufficient conditions for the origin to be a weakly persistent center, and the corresponding result for the associated general real cubic system follows by imposing the conjugacy relations between the complex coefficients. We further characterize the weakly persistent isochronous centers of the general complex cubic system by combining the weakly persistent center conditions with the vanishing of the linearization quantities and constructing time-preserving analytic linearizations.

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Cite

BibTeXRIS
BibTeX
@misc{li2026complete,
  title = {{Complete solution to the weak persistent center and isochronous center problems of cubic polynomial systems}},
  author = {Feng Li and Huimei Liu and Chen Cao and Pei Yu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.39761},
  eprint = {2609.39761},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.39761},
}
RIS
TY  - UNPB
TI  - Complete solution to the weak persistent center and isochronous center problems of cubic polynomial systems
AU  - Feng Li
AU  - Huimei Liu
AU  - Chen Cao
AU  - Pei Yu
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2609.39761
UR  - https://arxiv.org/abs/2609.39761
AB  - In this paper, we investigate weakly persistent centers for several classes of planar cubic differential systems, including systems with a nondegenerate linear center and systems with a nilpotent center. Necessary parameter relations are obtained by computing singular point quantities or quasi-Lyapunov constants, while their sufficiency is established by suitable analytic constructions. For the general complex cubic system, we obtain necessary and sufficient conditions for the origin to be a weakly persistent center, and the corresponding result for the associated general real cubic system follows by imposing the conjugacy relations between the complex coefficients. We further characterize the weakly persistent isochronous centers of the general complex cubic system by combining the weakly persistent center conditions with the vanishing of the linearization quantities and constructing time-preserving analytic linearizations.
ER  -