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PreprintarXiv licencearXiv

Some remarks on polynomial vector fields in RR2

  • Supriyo Jana
  • Soumen Sarkar
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Abstract

In this paper, we study planar polynomial vector fields admitting invariant circles. We characterize polynomial vector fields having m concentric invariant circles, and we describe those having three invariant circles whose centers are not collinear. We also give a necessary and sufficient condition for the unit circle to be the unique limit cycle of a quadratic vector field. We prove that a cubic or quartic Kolmogorov vector field having the unit circle invariant has no circular limit cycle, and that a quartic vector field with two concentric circular limit cycles has at most three circular limit cycles, this bound being sharp.

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Cite

BibTeXRIS
BibTeX
@misc{jana2026some,
  title = {{Some remarks on polynomial vector fields in $\RR^2$}},
  author = {Supriyo Jana and Soumen Sarkar},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.33226},
  eprint = {2610.33226},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.33226},
}
RIS
TY  - UNPB
TI  - Some remarks on polynomial vector fields in RR^2
AU  - Supriyo Jana
AU  - Soumen Sarkar
PY  - 2026
DA  - 2026/10/23
PB  - arXiv
DO  - 10.48550/arxiv.2610.33226
UR  - https://arxiv.org/abs/2610.33226
AB  - In this paper, we study planar polynomial vector fields admitting invariant circles. We characterize polynomial vector fields having m concentric invariant circles, and we describe those having three invariant circles whose centers are not collinear. We also give a necessary and sufficient condition for the unit circle to be the unique limit cycle of a quadratic vector field. We prove that a cubic or quartic Kolmogorov vector field having the unit circle invariant has no circular limit cycle, and that a quartic vector field with two concentric circular limit cycles has at most three circular limit cycles, this bound being sharp.
ER  -