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Cyclicity of lips and center-focus problem at infinity

  • Hebai Chen
  • Dehong Dai
  • Vadim Kaloshin
  • Zhijie Li
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Abstract

In this work, we discover an intriguing Liénard system which is integrable near infinity, integrable near zero and having Ilyashenko-Kotova lips in between. Moreover, we give a necessary and sufficient condition of polynomial Liénard system with center at infinity. In a different direction we improve a lower bound on Hilbert number for Liénard systems. The improvement is due to development of Brudnyi method of calculation of cyclicity of centers near infinity and near zero.

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Cite

BibTeXRIS
BibTeX
@misc{chen2026cyclicity,
  title = {{Cyclicity of lips and center-focus problem at infinity}},
  author = {Hebai Chen and Dehong Dai and Vadim Kaloshin and Zhijie Li},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.17773},
  eprint = {2608.17773},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.17773},
}
RIS
TY  - UNPB
TI  - Cyclicity of lips and center-focus problem at infinity
AU  - Hebai Chen
AU  - Dehong Dai
AU  - Vadim Kaloshin
AU  - Zhijie Li
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2608.17773
UR  - https://arxiv.org/abs/2608.17773
AB  - In this work, we discover an intriguing Liénard system which is integrable near infinity, integrable near zero and having Ilyashenko-Kotova lips in between. Moreover, we give a necessary and sufficient condition of polynomial Liénard system with center at infinity. In a different direction we improve a lower bound on Hilbert number for Liénard systems. The improvement is due to development of Brudnyi method of calculation of cyclicity of centers near infinity and near zero.
ER  -