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PreprintCC BY 4.0arXiv

A family of Hurwitz polynomial vector fields in ℝ3 having non-constant periodic orbits

  • Álvaro Castañeda
  • Salomón Rebollo-Perdomo
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Abstract

We construct a two-parameter family of polynomial vector fields in ℝ3 and prove that, on a nonempty open set of parameter values, every member of the family is Hurwitz and possesses two non-constant periodic orbits, one of saddle type and one attracting. This answers Problem 19 in Gasull's list of open problems on low-dimensional dynamical systems concerning the existence of non-constant periodic orbits in Hurwitz smooth vector fields on ℝ3. This also provides counterexamples to the expected non-existence of non-constant periodic orbits for Hurwitz polynomial vector fields; a question that arose from the first counterexamples to the Markus-Yamabe conjecture in the late 1990s and that was stated by Meisters in his featured MathSciNet review of the Bernat-Llibre counterexample.

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Cite

BibTeXRIS
BibTeX
@misc{castaneda2026family,
  title = {{A family of Hurwitz polynomial vector fields in $\mathbbR^3$ having non-constant periodic orbits}},
  author = {Álvaro Castañeda and Salomón Rebollo-Perdomo},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.09176},
  eprint = {2610.09176},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.09176},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A family of Hurwitz polynomial vector fields in ℝ^3 having non-constant periodic orbits
AU  - Álvaro Castañeda
AU  - Salomón Rebollo-Perdomo
PY  - 2026
DA  - 2026/10/06
PB  - arXiv
DO  - 10.48550/arxiv.2610.09176
UR  - https://arxiv.org/abs/2610.09176
AB  - We construct a two-parameter family of polynomial vector fields in ℝ^3 and prove that, on a nonempty open set of parameter values, every member of the family is Hurwitz and possesses two non-constant periodic orbits, one of saddle type and one attracting. This answers Problem 19 in Gasull's list of open problems on low-dimensional dynamical systems concerning the existence of non-constant periodic orbits in Hurwitz smooth vector fields on ℝ^3. This also provides counterexamples to the expected non-existence of non-constant periodic orbits for Hurwitz polynomial vector fields; a question that arose from the first counterexamples to the Markus-Yamabe conjecture in the late 1990s and that was stated by Meisters in his featured MathSciNet review of the Bernat-Llibre counterexample.
ER  -