Quartic polynomial differential systems with three disjoint invariant circles

@misc{cao2026quartic,
  title = {{Quartic polynomial differential systems with three disjoint invariant circles}},
  author = {Chen Cao and Jaume Llibre},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.05696},
  eprint = {2611.05696},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.05696},
}

TY  - UNPB
TI  - Quartic polynomial differential systems with three disjoint invariant circles
AU  - Chen Cao
AU  - Jaume Llibre
PY  - 2026
DA  - 2026/11/03
PB  - arXiv
DO  - 10.48550/arxiv.2611.05696
UR  - https://arxiv.org/abs/2611.05696
AB  - We determine the normal forms of real planar polynomial differential systems of degree at most four with three prescribed pairwise disjoint invariant circles, covering all configurations of their centers. None of the prescribed circles is a limit cycle in the concentric case or in the coaxial case with three distinct centers, where the pairwise radical axes coincide. When exactly two centers coincide, or when the three centers are distinct and collinear and the circles are non-coaxial, we give necessary and sufficient conditions for all three circles to be algebraic limit cycles, which are necessarily hyperbolic. For non-collinear centers, we give explicit necessary and sufficient conditions for all three circles to be periodic orbits and for all three to be hyperbolic algebraic limit cycles. Each of the four configurations with non-collinear centers is realized by a coprime quartic polynomial differential system with three hyperbolic circular limit cycles.
ER  -